SELECTIVE DEPOSITION MODELING METHOD AND APPARATUS FOR FORMING THREE-DIMENSIONAL OBJECTS AND SUPPORTS

Abstract: Methods of manipulating data in a thermal stereolithography apparatus, characterized in that the data represents a plurality of start/stop transitions to facilitate the computation of Boolean operations.

Description: SEE£JEi£AII Q N. SELECTIVE DEPOSITION MODELING METHOD AND APPARATUS FOR FORMING THREE- DIMENSIONAL OBJECTS AND SUPPORTS 1 Field of the Invention This invention relates to techniques for forming three-dimensional (3D) objects and supporting those objects dunng formation, more particularly, it relates to techniques for use in Rapid Prototyping and Manufacturing (RP&M) Systems, and most particularly to building and supporting methods and apparatus for use in a Thermal Stereolithography (TSL) system, Fused Deposition Modeling (FDM) system or other Selective Deposition Modeling (SDM) system 2 Background Information Various approaches to automated or semi-automated three-dimensional object production or Rapid Prototyping & Manufacturing have become available in recent years characterized in that each proceeds by building up 3D objects from 3D computer data descriptive of the objects in an additive manner from a plurality of formed and adhered laminae These laminae are sometimes called object cross-sections, layers of structure, object layers, layers of the object, or simply layers (if the context makes it clear that solidified structure of appropriate shape is being referred to) Each lamina represents a cross-section of the three-dimensional object Typically lamina are formed and adhered to a stack of previously formed and adhered laminae In some RP&M technologies, techniques have been proposed which deviate from a strict layer-by-layer build up process wherein only a portion of an initial lamina is formed and pπor to the formation of the remaining portιon(s) of the initial lamina, at least one subsequent lamina is at least partially formed According to one such approach, a three-dimensional object is built up by applying successive layers of unsolidified, flowable material to a working surface, and then selectively exposing the layers to synergistic stimulation in desired patterns, causing the layers to selectively harden into object laminae which adhere to previously-formed object laminae In this approach, material is applied to the working surface both to areas which will not become part of an object lamina, and to areas which will become part of an object lamina Typical of this approach is Stereolithography (SL), as descnbed in U S Patent No 4,575,330, to Hull According to one embodiment of Stereolithography, the synergistic stimulation is radiation from a UV laser, and the matenal is a photopolymer Another example of this approach is Selective Laser Sintenng (SLS), as descnbed in U S Patent No 4,863.538, to Deckard, in which the synergistic stimulation is IR radiation from a C0 2 laser and the material is a sinterable powder This first approach may be termed photo-based stereolithography A third example is Three- Dimensional Pπnting (3DP) and Direct Shell Production Casting (DSPC), as descnbed in U S Patent Nos 5,340,656 and 5,204,055 , to Sachs, et al , in which the synergistic stimulation is a chemical binder (e g an adhesive), and the matenal is a powder consisting of particles which bind together upon selective application of the chemical binder According to a second such approach, an object is formed by successively cutting object cross-sections having desired shapes and sizes out of sheets of matenal to form object lamina Typically in practice, the sheets of paper are stacked and adhered to previously cut sheets prior to their being cut, but cutting prior to stacking and adhesion is possible Typical of this approach is Laminated Object Manufactuπng (LOM), as descnbed in U S Patent No 4,752,352, to Feygin in which the mateπal is paper, and the means for cutting the sheets into the desired shapes and sizes is a C0 2 laser U.S Patent 5,015,312 to Kinzie also addresses building object with LOM techniques According to a third such approach, object laminae are formed by selectively depositing an unsolidified, flowable matenal onto a working surface in desired patterns in areas which will become part of an object laminae After or dunng selective deposition, the selectively deposited material is solidified to form a subsequent object lamina which is adhered to the previously-formed and stacked object laminae These steps are then repeated to successively build up the object lamιna-by-lamιna This object formation technique may be generically called Selective Deposition Modeling (SDM) The main difference between this approach and the first approach is that the material is deposited only in those areas which will become part of an object lamina Typical of this approach is Fused Deposition Modeling (FDM), as described in U S Patent Nos 5,121.329 and 5,340,433, to Crump, in which the material is dispensed in a flowable state into an environment which is at a temperature below the flowable temperature of the matenal, and which then hardens after being allowed to cool A second example is the technology descnbed in U S Patent No 5,260,009, to Penn A third example is Ballistic Particle Manufacturing (BPM), as described in U S Patent Nos 4 665,492, 5 134,569, and 5,216,616, to Masters, in which particles are directed to specific locations to form object cross- sections A fourth example is Thermal Stereolithography (TSL) as descnbed in U S Patent No 5,141 ,680, to Almquist et. al When using SDM (as well as other RP&M building techniques), the appropπateness of vanous methods and apparatus for production of useful objects depends on a number of factors As these factors cannot typically be optimized simultaneously, a selection of an appropnate building technique and associated method and apparatus involve trade offs depending on specific needs and circumstances Some factors to be considered may include 1 ) equipment cost, 2) operation cost, 3) production speed, 4) object accuracy, 5) object surface finish, 6) material properties of formed oojects, 7) anticipated use of objects, 8) availability of secondary processes for obtaining different material properties, 9) ease of use and operator constraints, 10) required or desired operation environment, 11 ) safety, and 12) post processing time and effort. In this regard there has been a long existing need to simultaneously optimize as many of these parameters as possible to more effectively build three-dimensional objects As a first example, there has been a need to enhance object production speed when building objects using the third approach, SDM, as descnbed above (e g Thermal Stereolithography) while simultaneously maintaining or reducing the equipment cost As a second example, there has been a long existing need for a low cost RP&M system useable in an office environment In SDM, as well as the other RP&M approaches, typically accurate formation and placement of working surfaces are required so that outward facing cross-sectional regions can be accurately formed and placed. The first two approaches naturally supply working surfaces on which subsequent layers of mateπal can be placed and lamina formed. However, since tne third approach, SDM, does not necessarily supply a working surface, it suffers from a particularly acute problem of accurately forming and placing subsequent lamina which contain regions not fully supported by previously dispensed mateπal such as regions including outward facing surfaces of the object in the direction of the previously dispensed mateπal. In the typical building process where subsequent laminae are placed above previously formed laminae this is particulariy a problem for down-facing surfaces (down- facing portions of laminae) of the object This can be understood by consideππg that the third approach theoretically only deposits material in those areas of the working surface which will become part of the corresponding object lamina Thus, nothing will be available to provide a working surface for or to support any down-facing surfaces appearing on a subsequent cross-section. Downward facing regions, as well as upward facing and continuing cross-sectional regions, as related to photo- based Stereolithography, but as applicable to other RP&M technologies including SDM are descnbed in detail in U S Patent Nos. 5,345.391 , and 5.321,622, to Hull et. al and Snead et. al , respectively The previous lamina is non-existent in down-facing regions and is thus unavailable to perform the desired support function Similarly, unsolidified material is not available to perform the support function since, by definition, in the third approach, such mateπal is typically not deposited in areas which do not become part of an object cross-section The problem resulting from this situation may be referred to as the "lack of working surface" problem The "lack of working surface" problem is illustrated in Figure 1, which depicts two laminae , identified with numerals 1 and 2, built using a three-dimensional modeling method and apparatus. As shown, lamina 1 , which is situated on top of lamina 2, has two down-facing surfaces, which are shown with cross-hatch and identified with numerals 3 and 4 Employing the SDM approach described above, unsolidified material is never deposited in the volumes directly below the down-facing surfaces, which are identified with numerals 5 and 6 Thus, with the SDM approach, there is nothing to provide a working surface for or to support the two down-facing surfaces Several mechanisms have been proposed to address this problem, but heretofore, none has proven completely satisfactory One such mechanism, suggested or descnbed in U S Patent No 4,247,508, to Housholder; U.S. Patent Nos. 4,961,154; 5,031 ,120, 5,263,130, and 5.386,500, to Pomeraπtz, et al.; U.S. Patent No. 5,136,515, to Helinski; U S. Patent No. 5,141 ,680, to Almquist, et al.; U.S Patent No. 5,260,009, to Penn; U.S. Patent No. 5,287,435, to Cohen, et al.. U.S. Patent No. 5,362,427, to Mitchell; U.S. Patent No 5,398,193, to dunghills; U.S. Patent Nos. 5,286,573 and 5,301 ,415, to Pππz, et al., involves filling the volumes below down-facing surfaces with a support mateπal different from that used to build the object, and presumably easily separable from it (by means of having a lower melting point, for example). In relation to Figure 1 , for example, the volumes identified with numerals 5 and 6 would be filled with the support mateπal prior to the time that the mateπal used to form down-facing surfaces 3 and 4 is deposited. A problem with the two mateπal (i e. building mateπal and different support material) approach is that it is expensive and cumbersome because of the inefficiencies, heat dissipation requirements, and costs associated with handling and delivering the support, or second, material For example, a separate mateπal handling and dispensing mechanism for the support mateπal may have to be provided Alternatively, means may have to be provided to coordinate the handling and delivery of both matenals through a single system Another approach, described in U S Patent No 4 999,143, to Hull, et al , U S Patent No 5,216,616, to Masters, and U S Patent No 5,386,500, to Pomerantz, et al , is to build generally spaced support structures from the same matenal as that used to build the object A multitude of problems have occurred with this approach A first problem has involved the inability to make support structures of arbitrary height while simultaneously ensuπng that they were easily separately from the object Second, a problem has been encountered regarding the inability to achieve easy separability between object and support structure while simultaneously maintaining an effective working surface for the building of and support of the outward facing surfaces A third problem involves the inability to accumulate support structure in the direction perpendicular to the planes of the cross-sections (e g vertical direction) at approximately the same rate as that at which the object accumulates A fourth problem has involved the inability to ensure easy separability and minimal damage to up-facing surfaces when supports must be placed thereon in order to support down-facing surfaces thereabove which are part of subsequent layers A fifth issue has involved the desire to increase system throughput To illustrate, the objective of achieving easy separability dictates that the surface area over which each support contacts the object be kept as small as possible On the other hand, the objective of accumulating a support in the Z-direction at a rate approximating that of object accumulation dictates that the cross-sectional area of each support be as large as possible to provide a large area to peπmeter ratio thereby minimizing loss of material for build up in the Z-direction due to run off, spreading, mis-targeting and the like by allowing a large target area to compensate for any inaccuracies in the deposition process and to limit the ability of material to spread horizontally instead of building up vertically Further, the objective of achieving minimal damage to the down-facing surface dictates that the spacing between the supports be kept as large as possible in order to minimize the area of contact between the supports and the object On the other hand, the objective of providing an effective working surface for the building of the down-facing surface dictates that the spacing be kept as small as possible As is apparent, there is a conflict in simultaneously achieving these objectives This problem is illustrated in Figure 2, in which, compared to Figure 1 , like elements are referenced with like numerals As shown down-facing surface 3 is supported through columnar supports 7a, 7b, and 7c, while down-facing surface 4 is supported through columnar supports 8a, 8b, 8c, and 8d. Columnar supports 7a, 7b, and 7c are widely spaced from one another in order to minimize damage to down-facing surface 3 Moreover, they are each configured to contact the down- facing surface over a relatively small surface area to enhance separability On the other hand, because of their small cross-sectional surface area, they may not be able to accumulate, in the vertical direction, fast enough to keep up with the rate of growth of the object Moreover, because of their wide spacing, they may not be able to provide an effective working surface for the building of and support of down-facing surface 3. Columnar supports 8a, 8b, 8c, and 8d, by contrast, are more closely spaced together in order to provide a more effective working surface for the building and support of down-facing surface 4. Also, each is configured with a larger surface area to enable them to grow at rate approximating that of the object Unfortunately, because of their closer spacing and larger cross-sectional area, these supports will cause more damage to the down-facing surface when they are removed All patents referred to herein above in this section of the specification are hereby incorporated by reference as if set forth in full 3 Attached Appendices and Related Patents and Applications Appendix A is attached hereto and provides details of preferred Thermal Stereolithography materials for use in the some preferred embodiments of the invention Appendix B is attached hereto and provides details of preferred data manipulation and system control techniques in a preferred SDM system. This is a copy of U.S. Patent Application Serial No. 08/ , , filed concurrently herewith, and corresponding to 3D System Docket No. USA 143. The following applications are hereby incorporated herein by reference as if set forth in full herein- Filing Date Application Title Status No. 9/27/95 08/534,813 Selective Deposition Modeling Pending Method and Apparatus for Forming Three-dimensional Objects and Supports 9/27/95 08/534.447 Method and Apparatus for Data Pending Manipulation and System Control in a Selective Deposition Modeling System 9/27/95 08/535,772 Selective Deposition Modeling Pending Materials and Method 9/27/95 08/534,477 Selective Deposition Modeling Pending Method and System According to Thermal Stereolithography and some Fused Deposition Modeling techniques, a three-dimensional object is built up layer by layer from a material which is heated until it is flowable, and which is then dispensed with a dispenser The material may be dispensed as a semi-continuous flow of material from the dispenser or it may alternatively be dispensed as individual droplets. In the case where the material is dispensed as a semi-continuous flow, it is conceivable that less stπngent working surface cπteria may be acceptable. An early embodiment of Thermal Stereolithography is described in U.S. Patent No. 5,141 ,680 Thermal Stereolithography is particularly suitable for use in an office environment because of its ability to use non-reactive, non-toxic materials Moreover, the process of forming objects using these mateπals need not involve the use of radiations (e g UV radiation, IR radiation, visible light and/or other forms of laser radiation), heating materials to combustible temperatures (e g burning the material along cross-section boundaries as in some LOM techniques), reactive chemicals (e g monomers, photopolymers) or toxic chemicals (e g solvents), complicated cutting machinery, and the like, which can be noisy or pose a significant risks if mishandled Instead, object formation is achieved by heating the material to a flowable temperature then selectively dispensing the material and allowing it to cool U S Patent Application No 08/534,447 referenced above, is directed to data transformation techniques for use in converting 3D object data into support and object data for use in a preferred Selective Deposition Modeling (SDM) system based on SDM/TSL principles This referenced application is also directed to various data handling data control, and system control techniques for controlling the preferred SDM/TSL system described hereinafter Some alternative data manipulation techniques and control techniques are also described for use in SDM systems as well as for use in other RP&M systems U S Patent Application No 08/535 772, as referenced above, is directed to the preferred mateπal used by the preferred SDM/TSL system described herein Some alternative materials and methods are also described U S Patent Application No 08/534,477, as referenced above is directed to some particulars of the preferred SDM/TSL system Some alternative configurations are also addressed The assignee of the instant application, 3D Systems, Inc , is also the owner of a number of other U S Patent Applications and U S Patents in RP&M field and particularly in the photo-based Stereolithography portion of that field These patents include disclosures which can be combined with the teachings of the instant application to provide enhanced SDM object formation techniques The following commonly owned U S Patent Applications and U S Patents are hereby incorporated by reference as if set forth in full herein App No Topic Status and/or Patent No 08/484,582 Fundamental elements of Stereolithography are taught Pending (DIV of 4,575,330 08/475,715 Vaπous recoating techniques for use in SL are described Pending (DIV of including a matenal dispenser that allows for selective 5,358,673) deposition from a plurality of oπfices 08/479,875 Various LOM type building techniques are described Pending (DIV of 5,192,559) 08/486,098 A descπption of curl distortion is provided along with vaπous Pending (DIV of techniques for reducing this distortion 5.104,592) 8/475,730 A description of a 3D data slicing technique for obtaining Pending (DIV of cross-sectional data is described which utilizes boolean 5,192,469) layer comparisons to define down-facing, up-facing and continuing regions Techniques for performing cure-width compensation and for producing vanous object configurations relative to an initial CAD design are also described 8/480,670 A description of an early SL Slicing technique is described Pending (DIV of including vector generation and cure width compensation 5,184.307) 08/428,950 Various building techniques for use in SL are described Pending (Parent for including various build styles involving alternate sequencing, W0 95/29053) vector interlacing and vector offsetting for forming semi-solid and solid objects 08/428,951 Simultaneously Multiple Layer Curing techniques for SL are Pending (Parent for taught including techniques for performing vertical W0 95/29053) comparisons, correcting errors due to over curing in the z- direction, techniques for performing horizontal comparisons, and horizontal erosion routines 08/405.812 SL recoating techniques using vibrational energy are Pending descnbed 08/402,553 SL recoating techniques using a doctor blade and liquid Pending (DIV of level control techniques are described. 5,174,931) 08/382.268 Several SL recoating techniques are described including Pending (Parent for techniques involving the use of ink jets to selectively PCT Application dispense material for forming a next layer of unsolidified No. US96/01451 matenal. 08/148,544 Fundamental elements of thermal stereolithography are 5,501 ,824 described. 07/182,801 Support structures for SL are described. 4,999,143 07/183,015 Placement of holes in objects for reducing stress in SL 5,015.424 objects are described 07/365,444 Integrated SL building, cleaning and post curing techniques 5,143,663 are described. 07/824,819 Various aspects of a large SL apparatus are described. 5,182,715 07/605,979 Techniques for enhancing surface finish of SL objects are 5,209,878 described including the use of thin fill layers in combination with thicker structural layers and meniscus smoothing. 07/929,463 Powder coating techniques are described for enhancing 5,234,636 surface finish. 07/939,549 Building techniques for reducing curl distortion in SL (by 5,238.639 balancing regions of stress and shπnkage) are described. SUMMARY OF THE INVENTION The instant invention embodies a number of techniques (methods and apparatus) that can be used alone or in combination to address a number of problems associated with building and supporting objects formed using Selective Deposition Modeling techniques. Though primarily directed to SDM techniques, the techniques described hereinafter can be applied in a variety of ways (as will be apparent to one of skill in the art who reads the instant disclosure) to the other RP&M technologies as described above to enhance object accuracy, surface finish, build time and/or post processing effort and time Furthermore, the techniques described herein can be applied to Selective Deposition Modeling systems that use one or more building and/or support materials wherein one or more are selectively dispensed and in which others may be dispensed non-selectively and wherein elevated temperatures may or may not be used for all or part of the materials to aid in their deposition The techniques can be applied to SDM systems wherein the building material (e.g. paint or ink) is made flowable for dispensing purposes by adding a solvent (e.g. water, alcohol, acetone, paint thinner, or other solvents appropriate for specific building, wherein the material is solidifiable after or during dispensing by causing the removal of the solvent (e.g. by heating the dispensed material, by dispensing the material into a partially evacuated (i e vacuumed) building chamber, or by simply allowing sufficient time for the solvent to evaporate) Alternatively, or additionally, the building material (e.g. paint) may be thixotropic in nature wherein an increase in shear force on the material could be used to aid in its dispensing or the thixotropic property may simply be used to aid the material in holding its shape after being dispensed Alternatively, and/or additionally, the material may be reactive in nature (e.g. a photopolymer. thermal polymer, one or two-part epoxy material, a combination mateπal such as one of the mentioned materials in combination with a wax or thermal plastic material) or at least solidifiable when combined with another material (e.g. plaster of pans & water), wherein after dispensing, the matenal is reacted by appropπate application of prescribed stimulation (e.g. heat, EM radiation [visible, IR, UV, x-rays, etc.], a reactive chemical, the second part of a two part epoxy, the second or multiple part of a combination) such that the building matenal and/or combination of materials become solidified. Of course, Thermal Stereolithographic materials and dispensing techniques may be used alone or in combination with the above alternatives. Furthermore, various dispensing techniques may be used such as dispensing by single or multiple ink jet devices including, but not limited to, hot melt ink jets, bubble jets, etc.. and continuous or semi-continuous flow, single or multiple orifice extrusion nozzles or heads. Accordingly it is a first object of the invention to provide a method and apparatus for higher accuracy production of objects. A second object of the invention is to provide a method and apparatus for production of objects with less distortion by controlling the thermal environment during object formation A third object of the invention is to provide a method and apparatus for production of objects with less distortion by controlling how material is dispensed. A fourth object of the invention is to provide a method and apparatus for enhancing object production speed. A fifth object of the invention is to provide a support structure method and apparatus that allows object supports of arbitrary height to be formed. A sixth object of the invenφon is to provide a support structure method and apparatus that provides a good working surface A seventh object of the invention is to a provide a method and apparatus that forms a support structure that is easily removed from down-facing surfaces of the object An eighth object of the invention is to provide a support structure method and apparatus that results in minimum damage to down-facing surfaces of the object upon removal thereof A ninth object of the invention is to provide a method and apparatus for removing the supports from the object A tenth object of the invention is to provide a support structure method and apparatus that builds up supports vertically at a rate approximating the vertical build up rate of the object An eleventh object of the invention is to provide a method and apparatus that forms a support structure that is easily removed from up-facing surfaces of the object A twelfth object of the invention is to provide a support structure method and apparatus that results in minimum damage to up-facing surfaces of the object upon removal thereof A thirteenth object of the invention is to provide a method and apparatus for producing supports that are separated from vertical object surfaces A fourteenth object is to provide support structures that are combmable with other RP&M technologies for enhanced object formation It is intended that the above objects can be achieved separately by different aspects of the invention and that additional objects of the invention will involve various combinations of the above independent objects such that combined benefits may be obtained from combined techniques Other objects of the invention will be apparent from the description herein BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 illustrates down-facing surfaces of an object, Figure 2 illustrates two categories of support structures for supporting the down-facing surfaces of Figure 1 , Figure 3 is a diagram of the main functional components of the preferred Selective Deposition Modeling/Thermal Stereolithography system, Figure 4a and 4b illustrate the orifice plate of the print head of Figure 3 at different orientations to the scanning direction, Figure 5 is a more detailed drawing of the planaπzer of Figure 3, Figure 6 illustrates the relative spacing between adjacent nozzles on the oπfice plate and adjacent raster lines, Figure 7 illustrates the grid of pixels which defines the data resolution of the system, Figure 8 illustrates two peφendicular examples of raster line orientation, Figure 9 illustrates two examples of deposition propagation in the secondary scanning direction, Figures 10a and 10b illustrates two examples of deposition propagation in (he main scanning direction; Figures 11a and 1 b illustrate an example of scan line interlacing; Figures 12a and 12b illustrate an example of drop location interlacing along several scan Iines; Figures 13a and 13b illustrate a further example of drop location interlacing along several scan lines; Figure 14 illustrates a single pixel checkerboard deposition pattern; Figure 15 illustrates a 3 x 3 column support pixel pattem forming a preferred support structure; Figures 16a - 16d illustrates several overpπnting schemes; Figures 17a and 17b illustrates a mis-registration problem that can occur when using an overpπnting technique; Figure 18 illustrates the resulting deposition regions when the pixels of Figure 15 are exposed using an overpπnting scheme. Figure 19 illustrates an alternative pixel pattem for column supports: Figure 20 illustrates a Hybrid support structure. Figures 21a and 21 illustrate arch-type supports: Figures 22a-d depict an interlacing embodiment for depositing material during the building of an object. Figures 23a-h illustrate a building embodiment which uses honzontal and vertical pixel offsets; Figures 24a-d illustrates a deposition embodiment that reduces risk of bridging between regions separated by a gap; Figures 25a-e illustrates a building technique wherein the object is separated into pieces, built separately and then adhered together: Figure 26 illustrates a preferred two step raster scanning and indexing pattern; Figures 27a-27e depict various combinations of working surface and targeting positions. Figure 28a depicts a side view of an embodiment of branching supports; Figure 28b depicts a side view of another embodiment of branching supports' Figures 29a-29e depict a top view of branching layers for an embodiment of branching supports; Figures 30a-30m depict a top view of branching layers for another embodiment of branching supports: Figures 31a-31c depict a top view of branching layers for another embodiment of branching supports: and Figures 32a-31 depict a top view of branching layers for another embodiment of branching supports DESCRIPTION OF THE PREFERRED EMBODIMENTS As previously discussed, the subject application is directed to support techniques and building techniques appropnate for use in a Selective Deposition Modeling (SDM) system. In particular, the preferred SDM system is a Thermal Stereolithography (TSL) system. The Description Of The Preferred Embodiments will begin with a descπption of the preferred TSL system A more detailed descπption of the preferred system, data manipulation techniques, system control techniques, mateπal formulations and properties, and various alternatives are descnbed in previously referenced and incoφorated U.S Patent Application Nos 08/534,813, 08/534,447, 08/535,772, and 08/534,477, and 3D Docket Nos. USA 143, filed concurrently herewith Further, alternative systems are discussed in a number of the previously incorporated applications and patents, especially those referenced as being directly related to, or applicable to, SDM, TSL or Fused Deposition Modeling (FDM) As such, the support structures and build styles descnbed hereinafter should be construed as applicable to a vaπety of SDM, TSL and FDM systems and not limited by the system examples descnbed herein Furthermore, as noted previously, these support structures and build styles have utility in the other RP&M technologies A preferred embodiment of an apparatus for performing SDMTSL is illustrated in Figure 3 The apparatus comprises a dispensing platform 18 a dispensing head 9 (e g multi orifice ink jet head), wherein the dispensing head 9 is located on the dispensing platform 18 a planaπzer 11 and a part-building platform 15 The dispensing platform 18 is a horizontal member which is capable of supporting the olanaπzer 11 and the dispensing head 9 The dispensing platform 18 is slidably coupled to an X-stage 12 through a coupling member 13 The X-stage 12 is preferably controlled by a control computer or microprocessor (not shown) and controUably moves the dispensing platform 18 back and forth in the X-direction, or the main scanning direction Furthermore, at either side of the platform 18, fans (not shown) are mounted for blowing air vertically downward to help cool the dispensed material 14 and part-building platform 15 such that the desired building temperature is maintained Other suitable mounting schemes for the fans and/or other cooling systems include, but are not limited to, misting devices for directing vaporizable liquids (e g water, alcohol, or solvents) onto the surface of the object, forced air cooling devices with fans mounted between the planaπzer 11 and the dispensing head 9, and forced air cooling devices with stationary or moving fans mounted off the dispensing platform Cooling systems may include active or passive techniques for removing heat which may be computer controlled in combination with temperature sensing devices to maintain the previously dispensed material within the desired building temperature range. Other approaches to cooling include, but are not limited to, salting the mateπal with a substance which functions as a black body radiator, especially at IR frequencies, such that heat is more readily radiated from the object during the building process Further alternative approaches include, but are not limited to, adding a conductive substance to the material every few layers, adding a solvent to the mateπal, building parts with cooling passages or with an embedded substrate (such as interlaced wires) for cooling, or building on a glass plate or mylar sheet Other embodiments for cooling the mateπal or at least maintaining the dispensed material at an appropriate temperature, might involve the use of directing a temperature moderating gas (e.g. a cooling gas such as air) at the upper surface of the partially formed object, as discussed above, but they may additionally include controlled techniques for removing the cooling air from the surface Such techniques might involve the use of blowing and sucking devices and altemate positioning of blowing ducts (gas inserting ducts) and sucking ducts (gas removing ducts) These ducts may allow the cooling gas to be removed before excessive heating of the gas causes a loss in effective cooling rate The gas directed at the surface may be introduced in a cooled state, introduced at room temperature, or introduced at some other appropriate temperature If appropπately configured, these alternate inserting and removing ducts may allow faster scanning speed than presently allowable due to turbulence or wind distortion of fragile structures such as supports These ducts might be configured to provide air flow in the opposite direction to print head movement thereby reducing the net wind velocity coming into contact with the partially formed object The blowing or sucking associated with individual ducts may be reversed turned on or tumed off depending on the direction of movement of the print head The print head 9 is a commercial print head configured for jetting hot melt inks such as, for example thermal plastics or wax-like materials, and modified for use in a three-dimensional modeling system wnerem the print head undergoes back and forth movements and accelerations The pπnt head modifications include configuring any on board reservoir such that the accelerations result in minimal misplacement of material in the reservoir One preferred embodiment includes a 96 jet commercial print head, Model No HDS 96ι sold by Spectra Corporation. Nashua, Hew Hampshire including reservoir modifications The print head is supplied material in a flowable state from a Material Packaging & Handling Subsystem (not shown) which is descnbed in the previously referenced U S Patent Application No 08/534,477 In the preferred embodiment, all 96 jets on the head are computer controlled to selectively fire droplets through orifice plate 10 when each orifice (i e jet) is appropriately located to dispense droplets onto desired locations In practice, approximately 12,000 to 16 000 commands per second have been sent to each jet selectively commanding each one to fire (dispense a droplet) or not to fire (not to dispense a droplet) depending on jet position and desired locations for material deposition Also in practice, firing commands have been sent simultaneously to all jets Since, the preferred print head mentioned above contains almost 100 jets, the above noted firing rates result in the need to send approximately 1 2 to 1 6 million firing commands to the head each second Thus, the head is computer controlled so as to selectively fire the jets and cause them to simultaneously emit droplets of the molten mateπal through one or more orifices in oπfice plate 10 Of course, it will be appreciated that in alternative preferred embodiments, heads with different numbers of jets can be used, different firing frequencies are possible, and in appropriate circumstances non-simultaneous firing of the jets is possible To most effectively build a three-dimensional object, it is desired that all of the jets first correctly To ensure that all jets are firing correctly or at least maximize the number which are firing correctly, vaπous techniques can be used One such embodiment involves checking the jets after formation of each lamina This technique includes the steps of 1 ) forming a lamina , 2) checking the jets by pπnting a test pattem of lines on a piece of paper, with all jets firing, 3) optically detecting (through bar code scanning or the like) whether a jet is misfiπng, 4) uncloggmg the jet, 5) removing the entirety of the just-dispensed layer (e g by machining using a preferred planaπzer to be described herein after), and 6) rebuilding the layer with all jets including the unclogged jet A second embodiment involves the following preferred steps 1 ) forming a layer, 2) optically detecting a misfiπng jet, 3) rescanning the lines on the layer that should have been formed by the misfiπng jet, 4) ceasing the use of the misfiring jet in the remainder of the building process, and 5) scanning subsequent layers while compensating for the misfiring jet (i e make extra passes with a working jet to cover the lines corresponding to the misfiring jet) Optionally, the misfiπng jet may be penodically checked to see if it has started functioning again If so this jet is put back into operation Another option involves putting a misfiπng jet through a reactivation routine to see if it can be made operational This could occur during the building process or during servicing of the system As a further alternative it may be possible to determine whether or not a jet is firing correctly by tracking the electπcal charactenstics of the piezo electric element as firing is to occur A third embodiment might involve the use of a flexible element for wiping excess material from the bottom of the pππt head This embodiment involves the firing of all the jets followed by a wiping of the orifice plate with a heated rubber ie g VITON) blade Preferably the blade is positioned such that it contacts the orifice plate as they are relatively moved passed each other thereby causing a squeegee action to remove excess material from the orifice plate and hopefully revitalizing any jets which were not behaving properly It is further preferred that the orifice plate and blade be positioned at an angle to each other such that at any one time during their contact only a portion of the orifice plate is in contact with the squeegee thereby minimizing the force the blade exerts on the orifice plate The orifice plate 10 is mounted on the dispensing platform 18 such that droplets of material are allowed to emit from the underside of the dispensing platform 18 The oπfice plate 10 is illustrated in Figures 4a and 4b In one preferred embodiment, and as depicted in Figure 4a, the orifice plate 10 (i e the line of orifices) is mounted approximately peφendicular to the main scanning direction (X- direction) and is configured with N = 96 individually controllable orifices (labeled 10(1 ), 10(2) 10(3) 10(96))Each oπfice is equipped with a piezoelectric element that causes a pressure wave to propagate through the material when an electric firing pulse is applied to the element The pressure wave causes a drop of material to be emitted from the orifice The 96 orifices are controlled by the control computer which controls the rate and timing of the firing pulses applied to the individual orifices With reference to Figure 4a the distance 'd" between adjacent orifices in the preferred embodiment is about 8/300 of an inch (about 26 67 mils or 0 677 mm) Thus, with 96 orifices the effective length "D" of the orifice plate is about (N x 8/300 inch) = (96 x 8/300 inches) = 2 56 inches (65 02 mm) To accurately build an object the print head must fire such that the droplets reach particular "desired drop locations", i e , locations that the droplet is intended to land The desired drop locations are determined from a data map, or pixel locations which descπbes the object as a series of relatively spaced location points For the droplets to land at the desired drop locations the print head must fire tie droplets from a "desired finng location" or at a "desired finng time" which is based on the relative position of the print head to the desired drop location, the velocity of the print head, and the ballistic charactenstics of the particles after being ejected In a preferred embodiment raster scanning is used to position the print head 9 and oπfices at desired finng locations The printing process for each lamina is accomplished by a seπes of relative movements between the head 9 and the desired drop or finng locations Printing typically occurs as the head 9 relatively moves in a main scanning direction This is followed by a typically smaller increment of movement in a secondary scanning direction while no dispensing occurs, which in turn is followed by a reverse scan in the main scanning direction in which dispensing again occurs The process of alternating main scans and secondary scans occurs repeatedly until the lamina is completely deposited Alternative preferred embodiments may perform small secondary scanning movements while main scanning occurs Because of the typically large difference in net scanning speed along the main and secondary directions such an alternative still results in deposition along scanning lines which are nearly parallel to the mam scanning direction and peφendicular to the secondary scanning direction Further alternative preferred embodiments may utilize vector scanning techniques or a combination of vector scanning and raster scanning techniques It has been found that droplets immediately after being dispensed from the jet orifice, have an elongated shape, compared to their width The ratio of droplet length to width can be defined as the aspect ratio of the αroplet It has further been found that the aspect ratio of these droplets becomes smaller as the droplets travel away from the jet orifice (i e they become more spherical in shape) It should be appreciated that in some embodiments the spacing between the oπfice plate 10 and the working surface is preferably large enough such that the drops emitted therefrom have become semi-circular in shape when they impact the working surface On the other hand, it should also be appreciated that this spacing, which determines the distance the droplets must travel during the printing process before impact, should be minimized in order to avoid accuracy problems which may occur as the travel time is increased In practice, it has been found that these two conditions are both satisfactorily met when at least 90% of the droplets emitted from the orifice plate have achieved an aspect ratio (i e , the ratio formed by the width of the droplet divided by its length) which is preferably less than about 1 3 more preferably less than about 1 2 and most preferably, which is between about 1 05 and 1 1 In alternative preferred embodiments, the print head 9 may be mounted at a non- pβφendicular angle to the main scanning direction This situation is depicted in Figure 4b wherein the print head 9 is mounted at an angle "α ' to the main scanning direction (e g , the "X" direction) In this alternative situation the separation between the orifices is reduced from d to d' = (d x sin α) and the effective length of the print head 9 is reduced to D' = (D x sin a) When the spacing d' is equal to the desired pπnt resolution in the secondary scanning direction (direction approximately peφβndicular to the main scanning direction), the angle α is considered to be the "saber angle" If the spacing d or d' is not at the desired secondary pπnt resolution (i e the print head is not at the saber angle) then for optimal efficiency in printing a layer, the desired resolution must be selected such as to make d or d' an integer multiple of the desired resolution Similarly, when printing with α *90°, a spacing between jets exists in the main scanning direction as well as the secondary scanning direction This spacing is defined by d"= d x cos σ This in turn dictates that optimization of printing efficiency will occur when the desired main direction pπnt resolution is selected to be an integral divisor of d" (this assumes that firing locations are located in a rectangular grid). Another way of expressing this is that the angle α is selected such that d' and/or d" when divided by appropriate integers M and P yield the desired secondary and main scanning resolutions. An advantage to using the preferred print head orientation (α = 90°) is that it allows any desired printing resolution in the main scanning direction while still maintaining optimal efficiency. In other preferred embodiments, multiple heads may be used which lay end to end (extend in the secondary scanning direction) and/or which are stacked back to back (stacked in the main scanning direction) When stacked back to back, the print heads may have orifices aligned in the main scanning direction so that they print over the same lines or alternatively they may be offset from one another so as to dispense material along different main scanning lines. In particular, it may be desirable to have the back to back print heads offset from each other in the secondary scanning direction by the desired raster line spacing to minimize the number of main scanning passes that must occur In other preferred embodiments, the data defining deposition locations may not be located by pixels configured in a rectangular grid but instead may be located by pixels configured in some other pattern (e.g offset or staggered pattern) More particularly, the deposition locations may be fully or partially varied from layer to layer in order to perform partial pixel drop location offsetting for an entire layer or for a portion of a layer based on the particulars of a region to be jetted Presently preferred printing techniques involve deposition resolutions of 300, 600 and 1200 drops per inch in the main scanning direction and 300 drops per inch in the secondary scanning direction With reference to Figures 3 and 5, the planaπzer 11 includes a heated rotating (e.g 2000 rpm) cylinder 18a with a textured (e.g. knurled) surface. Its function is to melt, transfer and remove portions of the previously dispensed layer or lamina of mateπal in order to smooth it out, to set a desired thickness for the last formed layer, and to set the net upper surface of the last formed layer to a desired level. Numeral 19 identifies a layer of matenal which was just deposited by the print head. The rotating cylinder 18a is mounted in the dispensing platform such that it is allowed to project from the underside of the platform by a sufficient amount in the Z-direction such that it contacts material 19 at a desired level. More importantly the rotating cylinder 18a is mounted so as to project a desired distance below the plane swept out by the underside of the print head or orifice plate. In the event that the onfice plate itself projects below the dispensing platform 18, the rotating cylinder 18a will project further below the dispensing platform 18. In one preferred embodiment, the projection below the oπfice plate in the z-direction is in the range of 0.5 mm to 1.0 mm. The extent to which the roller extends below the dispensing platform 18 is a determinant of the spacing between the oπfice plate 10 and the working suiface. Thus, in some preferred embodiments it is preferred that the extent to which the planaπzer 11 extends below the orifice plate 10 not conflict with the condition descnbed earlier in relation to droplet aspect ratio, in which 90% of the droplets have achieved an aspect ratio upon impact preferably less than about 1.3, more preferably less than about 1.2, and most preferably between about 1.05-1.1 The rotation of the cylinder sweeps material from the just-deposited layer, identified in the figure with numeral 21, leaving in its wake smooth surface 20 The material 21 adheres to the knurled surface of the cylinder and is displaced until it contacts wiper 22 As shown wiper 22 is disposed to effectively "scrape" the material 21 from the surface of the cylinder The wiper is preferably made of VITON, although other materials such as TEFLON®, are capable of scraping the mateπal from the surface of the cylinder are also suitable Preferably the scrapper material is non-wetting with respect to the liquefied building material and is durable enough to contact the rotating cylinder 18a without wearing out too quickly The removed mateπal is drawn away under suction via a heated umbilical to a waste tank (not shown), where it is either disposed of or recycled The planaπzer waste tank is held constantly under vacuum in order to continuously remove material from the planaπzer cylinder When the tank becomes full the system automatically reverses the vacuum for a few seconds to purge the waste material out of a check valve into a larger waste tray Once empty, vacuum is restored and waste continues to be drawn from the planaπzer In practice, it has been observed that approximately 10-15% of the material dispensed is removed by the plananzer Though most preferred embodiments use a combination of rotating, melting and scraping to perform planarization it is believed that other embodiments might utilize any one of these three elements or any combination of two of them In present implementations the cylinder 18a rotates (e g at approximately 2000 rpm) in a single direction as the head moves back in forth in each direction In alternative embodiments, the cylinder 18a can be made to rotate in opposite directions based on the forward or reverse direction that platform 18 sweeps while moving in the main scanning direction Some embodiments might involve the axis of rotation of cylinder 18a being off axis relative to the axis of orientation of the print head In other embodiments more than one cylinder 18a may be used For example, if two cylinders were used, each one may be caused to rotate in different directions and may further be vertically positionable so as to allow a selected one to participate in planarization during any given sweep When using a single print head 10 and cylinder 18a, planarization only effectively occurs on every second pass of the print head though deposition occurs on each pass (i e planarization always occurs in the same direction) Under these conditions, planarization occurs when the sweeping direction points along the same direction as an arrow pointing from the cylinder to the print head In other words planaπzation occurs when the sweeping direction is such that the cylinder follows the pπnt head as the elements traverse the layer in the main scanning direction Other preferred embodiments might use a single cylinder, but use one or more pπnt heads located on either side of the cylinder, such that planarization effectively occurs when sweeping in both directions Other alternative embodiments might decouple the movement of the print head(s) and the planarizing cylinder This decoupling might allow independent plananzation and dispensing activity Such decoupling might involve the directions of print head sweeping (e g X-direction) and cylinder sweeping (e g Y-direction) being different Such decoupling might also allow multiple layers to be formed or lines of a single layer to be deposited between planarization steps With reference to Figure 3a, part-building platform 15 is also provided The three-dimensional object or part, identified in the figure with reference numeral 14, is built on the platform 15 The platform 15 is slidably coupled to Y-stage 16a and 16b which controUably moves the platform back and forth in the Y-direction (i e , index direction or secondary scanning direction) under computer control The platform 15 is also coupled to Z-stage 17 which controUably moves the platform up and down (typically progressively downward during the build process) in the Z-direction under computer control To build a cross-section lamina, or layer of a part the Z-stage is directed to move the part- building platform 15 relative to the print head 9 such that the last-built cross-section of the part 14 is situated an appropriate amount below the orifice plate 10 of the print head 9 The print head 9 in combination with the Y-stage 16a 16b is then caused to sweep one or more times over the XY build region (the head sweeps back and forth in the X direction while the Y-stage 16a, 16b translates the partially formed object in the Y-direction) The combination of the last formed cross-section lamina, or layer of the object and any supports associated therewith define the working surface for deposition of the next lamina and any supports associated therewith During translation in the XY directions, the jet orifices of the print head 9 are fired in a registered manner with respect to previously dispensed layers to deposit material in a desired pattern and sequence for the building of the next lamina of the object During the dispensing process a portion of the dispensed material is removed by the plananzer 11 in the manner discussed above The X Y and Z movements dispensing, and planarizing are repeated to build up the object from a plurality of selectively dispensed and adhered layers Moreover platform 15 can be indexed in either the Y- or Z- direction while the direction of the dispensing platform 18 is in the process of being reversed upon the completion of a scan In a preferred embodiment, the material deposited during the formation of a lamina has a thickness at or somewhat greater than the desired layer thickness As descnbed above the excess mateπal deposited is removed by the action of the plananzer Under these conditions, the actual build up thickness between layers is not determined by the amount of material deposited for each layer but instead is determined by the down-ward vertical increment made by the platform after deposition of each layer If one wants to optimize build speed and/or minimize the amount of mateπal wasted, it is desirable to trim off as little material as possible during the deposition process The less material tπmmed off the thicker each lamina is and the faster the object builds up On the other hand if one makes the layer thickness, i e z-increment too large then the amount of build up associated with at least some drop locations will begin to lag behind the desired level This lag will results in the actual physical working surface being at a different position from the desired working surface and probably results in the formation of a non-planar working surface This difference in position can result in the XY misplacement of droplets due to a longer time of flight for than expected and it can further result in the vertical misplacement of object features that happen to begin or end at the layers in which the actual working surface is mis-positioned Therefor in some embodiments it is desirable to optimize layer incrementing in the vertical direction To determine an optimum Z-axis increment, an accumulation diagnostic part may be used This technique preferably involves building layers of one or more test parts at successively greater Z- increments, measuπng the height of the features formed and determining which Z-increments gave rise to formation heights (i e vertical accumulation) of the correct amount and which Z-increments gave rise to formation heights which lagged behind the desired amount It is expected that layer- increments (i e Z-increments) up to a certain amount (i e the maximum acceptable amount) would yield build up levels for the object equal to that predicted by the product of the number of layers and the thickness of each layer After the layer increment exceeds the maximum acceptable amount, the build up level of the object would fall short of the amount predicted by the product of the number of layers and the thickness of each layer Alternatively, the planarity of upper surface of the diagnostic part(s) may be lost (indicating that some drop locations may be receiving sufficient mateπal while others are not) By inspecting the diagnostic part(s), the maximum acceptable Z-increment amount can be empirically determined The optimal Z-increment amount can then be selected as this maximum acceptable amount or it can be selected at some thickness somewhat less than this maximum amount Since it is known that different build and support styles accumulate in the vertical direction at different rates the above test can be performed for each build style and support style, wherefrom the optimal Z-increment for a combination of different styles can then be selected such that it is not thicker tnan any of the maximum amounts determined for each style individually Further, the dispensing head in tracing a given scan line, may only maintain a substantially constant velocity over part of the scan line During the remainder of the scan the head 9 will either be accelerating or decelerating Depending on how the firing of the jets is controlled this may or may not cause a problem with excess build up during the acceleration and deceleration phases of the motion In the event that velocity changes can cause problem in a accumulation rate, the part or support building can be confined to the portion of the scan line over which the print head has a substantially constant velocity Alternatively, as discussed in the concurrently filed U S Patent Application corresponding to 3D Docket No USA 143, a finng control scheme can be used which allows accurate deposition duπng the acceleration or deceleration portions of a scan line As noted previously, in some preferred embodiments, the print head 9 is directed to trace a raster pattem An example of this is depicted in Figure 6 As shown, the raster pattern consists of a senes of raster lines (or scan lines), R(1) R(2), , R(N), running in the X-direction or main scanning direction and arrayed (I e spaced) along the Y-direction (i e the index direction or secondary scanning direction) The raster lines are spaced from one another by a distance d r , which, in one preferred embodiment is 1/300 of an inch (about 3 3 mils or about 83 8 μm) Since the orifices of the print head 9 are spaced by the distance d, which as discussed above is preferably about 26 67 mils (0 6774 μm ), and since the desired number of raster lines may extend in the index direction by a distance greater than the length of the oπfice plate 10, about 2 56 inches (65 02 mm), the print head 9 must be swept over the working surface through multiple passes in order to trace out all desired raster lines This is preferably accomplished by following a two-step process In the first step, the pnnt head 9 is passed 8 times over the working surface in the ma scanning direction, with the Y-stage 16a, 16b being indexed by the amount d r in the secondary scanning direction after each pass in the main scanning direction In the second step, the Y-stage 16a 16b is indexed by a distance equal to the length of the oπfice plate 10 (2.5600 inches + d r (0.0267 inches) = 2 5867 inches (65 70 mm) This two-step process is then repeated until all of the desired raster lines have been traced In a first pass, for example, the print head 9 might be directed to trace raster lines R(1) ( via orifice 10(1) in Figure 4), R(9) ( via orifice 10(2)), R(17) ( via orifice 10(3)), etc The Y-stage 16a, 16b would then be directed to move the building platform 18 the distance d r (one raster line) in the index direction On the next pass, the print head 9 might be directed to trace R(2) (via 10(1)), R(10) (via 10(2)), R(17) (via 10(3)), etc Six more passes would then be performed with the Y-stage 16a, 16b being indexed by the distance d r after each pass, until a total of 8 passes have been performed After performing the first step (consisting of 8 passes), the second step is performed if there are more raster lines to be traced The second step consists of directing the Y-stage to move the building platform by an amount equal to the full length of the orifice plate 10 + d r , 2.5867 inches (65 70 mm) As needed, another set of 8 passes, comprising the first step, is performed followed by another second step The two-step process described above would then be repeated until all raster lines have been traced out An example of this two step process is depicted in Figure 26 for a pπnt head consisting of two jets and wherein the two jets are separated one from the other by 8 raster spacings The scanning of the cross-sections begins with the first jet located at position 201 and the second jet located at position 301 The first step of the scanning process begins with the scanning of raster lines 211 and 311 in the indicated direction by the first and second jets, respectively As part of the first step the initial scanning of raster lines 211 and 311 is followed by an index increment of one raster line width as indicated by elements 221 and 321 Continuing as part of the first step, the initial raster scan and index increment are followed by seven more raster scans (depicted by pairs of lines 212 and 312, 213 and 313, 214 and 314, 215 and 315, 216 and 316, 217 and 317, and 218 and 318) separated by six more 1 raster line width index increments (depicted with pairs of elements 222 and 322, 223 and 323, 224 and 324, 225 and 325, 226 and 326, and 227 and 327) Immediately after scanning raster line pairs 218 and 318, the second step of the process is taken wherein the head is indexed in the Y- direction according to the direction and lengths of raster lines 228 and 229 The length of this index is equal to the head width (i e in this example 8 raster lines widths) plus the width of 1 more raster line After this large increment, the first steps and second steps are repeated as many times as necessary to complete the scanning of the particular cross-section being formed It will be apparent to one of skill in the art that this two step scanning technique can be implemented in other ways in alternative embodiments. For example the second step may, instead of consisting of the positive index increment in Y as indicated by elements 228 and 328, consist of the large negative increment in Y as indicated by element 330 (i e three head widths minus one raster line width) This preferred embodiment may be summarized as including the following charactenstics 1) the spacing along an indexing direction between adjacent jets is an integral (N) multiple of the desired spacing (d r ) between adjacent deposition lines which extend in a pπnting direction which is approximately peφendicular to the indexing direction, 2) the first step includes performing a number of passes (N) in the pπnting direction where each pass is offset in the indexing direction by the desired spacing (d r ) between adjacent deposition lines; and 3) the second step includes offsetting the pπnt head 9 in the indexing direction by a large amount such that the jets can deposit mateπal in mateπal in another N passes, wherein successive passes are separated by one raster line index increments, and whereafter another large index increment will be made as necessary In most preferred embodiments the second step index amount will be equal to the sum of the spacing between the first jet and the last jet plus the desired spacing between adjacent deposition lines (i e , N x J + d„ where J is the number of jets on the print head 9) As noted in the above example other second step index amounts are possible For example, negative second step increments (opposite direction to the index increments used in the first step) equal to the sum of the head width plus the product of two times the width between successive jets less the width of one raster line spacing In other embodiments it is possible to use second step index amounts which vary or which alternate back and forth between positive and negative values In these embodiments the second step increment amount has the common feature that it is larger than the individual index amounts used in the first step In other preferred embodiments other single or multiple step indexing pattems can be used index direction increments could be generally be made which include increments involving both negative and positive movements along the Y-axis This might be done to scan raster lines that were initially skipped This will be described further in association with a technique called "interlacing In some preferred embodiments the firing of ink jets is controlled by a rectangular bit map, i e , pixel locations maintained in the control computer or other memory device The bit map consists of a grid of memory cells in which each memory cell corresponds to a pixel of the working surface, and in which the rows of the grid extend in the main scanning direction (X-direction) and the columns of the grid extend in the secondary scanning direction (Y-direction) The width of (or distance between) the rows (spacing along the Y-direction) may be different from the width (or length of or distance between ) of the columns (spacing along the X-direction) dictating that different data resolutions may exist along the X and Y directions In other preferred embodiments non-uniform pixel size is possible within a layer or between layers wherein one or both of the pixel width or length is varied by pixel position In still other preferred embodiments other pixel alignment patterns are possible For example, pixels on adjacent rows may be offset in the main scanning direction by a fractional amount of the spacing between pixels so that their center points do not align with the center points of the pixels in the neighboring rows This fractional amount may be Vi so that their center points are aligned with the pixel boundaries of adjacent rows It may be 1/3 1/4 or some other amount such that its takes two or more intermediate layers before pixel pattems realign on subsequent layers In further alternatives, pixel alignment might be dependent on the geometry of the object or support structure being dispensed For example it might be desirable to shift pixel alignment when forming a portion of a support pattem that is supposed to bπdge a gap between support columns or when forming a down-facing portion of an object These and other alternative pixel alignment schemes can be implemented by modifying the pixel configuration, or alternatively, defining a higher resolution pixel arrangement (in X and/or Y) and using pixel firing patterns that do not fire on every pixel location but instead fire on selected spaced pixel locations which may vary according to a desired random, predetermined or object biased pattern The data resolution in the main scanning direction can be defined in terms of Main Direction Pixels (MDPs) MDPs may be described in terms of pixel length or in terms of number of pixels per unit length In some preferred embodiments, MDP = 300 pixels per inch (26 67 mils/pixel or 6774 μm/pixel) In other preferred embodiments MDP = 1200 pixels per inch Of course any other MDP values can be used as desired Similarly the data resolution in the secondary scanning direction may be defined in terms of Secondary Direction Pixels (SDPs) and the SDPs may be described in terms of pixel width or in terms of number of pixels per unit length In some preferred embodiments SDP = MDP = 300 pixels per inch (26 67 mils/pixel or 677 4 μm/pixel) The SDP may or may not be equivalent to spacing between raster lines and the MDP may or may not be equivalent to the spacing between successive drop locations along each raster line The spacing between successive raster lines may be defined as Secondary Drop Locations (SDLs), while spacing between successive drop locations along each raster line may be defined as Main Drop Locations (MDLs) Similar to SDPs and MDPs, SDLs and MDLs may be defined in terms of drops per unit length or drop spacing If SDP = SDL there is a one to one correspondence between data and drop locations along the secondary scanning direction and the pixel spacing is equal to that of the raster line spacing If MDP = MDL there is a one to one correspondence between data and drop locations along the main scanning direction If SDL and/or MDL is larger than SDP and/or MDP respectively, more drops will need to be fired than that for which data exists thus each pixel will need to be used to control the dropping of more than one droplet The dispensing of these extra droplets can be done either by dispensing the droplets at intermediate points between the centers of successive pixels (i e intermediate dropping, "ID") or alternatively, directly on top of pixel centers (i e direct dropping, "DD") In either case this technique is called "overprinting' and results in faster build up of mateπal and eases mechanical design constraints involving maximum scan speeds and acceleration rates since the same Z-build up can occur while moving the pπnt head and/or object more slowly The difference in ID overprinting versus non-overpπnting, or DD overpπnting is depicted in Figures 16a to 16d Figure 16a depicts a single drop 60 being deposited and an associated solidified region 62 surrounding it when the print head is moving in direction 64 On the other hand, Figure 16b depicts the same region being cured but using the ID overprinting technique where two drops 60 and 66 are deposited in association with the single data point when the head is moving in direction 64 The deposition zone filled by the two drops is depicted by region 68 Figure 16c shows a similar situation for a four drop ID overpπnting scheme wherein the drops are indicated by numerals 60, 70, 66 and 72 and the deposition zone is depicted by 76 and wherein the scanning direction is still depicted by numeral 64 Figure 16d depicts a similar situation for a line of pixels 78 80 82, 84, 86 and 88 wherein numeral 90 depicts the length of the deposition zone without overprinting and the numeral 92 depicts the length of the deposition 7one when using a four drop ID overpnnting technique. The above can be generalized by saying that ID overpπnting adds from approximately V. to just under 1 additional pixel length to any region wherein it is used Of course, the more overprinting drops that are used, the more vertical growth a pixel region will have If SDL and/or MDL is less than SDP and/or MDP, respectively, drops will be fired at fewer locations than those for which data exists, at least for a given pass of the print head This data situation may be used to implement the offset pixel and/or non-uniform sized pixel techniques discussed above An N row by M column grid is depicted in Figure 7 As shown, the rows in the grid are labeled as R(1 ), R(2), , R(N), while the columns in the grid are labeled as C(1 ), C(2), , C(M) Also shown are the pixels making up the grid These are labeled as P(1 ,1 ), P(1 ,2), ,P(M,N) To build a cross-section, the bit map is first loaded with data representative of the desired cross-section (as well as any supports which are desired to be built) Assuming, as with some preferred embodiments, a single build and support material is being used If it is desired to deposit material at a given pixel location, then the memory cell corresponding to that location is appropriately flagged (e g loaded with a binary "1") and if no material is to be deposited an opposite flag is used (e g a binary "0") If multiple materials are used, cells corresponding to deposition sites are flagged appropπately to indicate not only drop location sites but also the material type to be deposited For ease of data handling, compressed data defining an object or support region (e g RLE data which defines on-off location points along each raster line as described in concurrently filed U S Patent Application No corresponding to 3D Systems Docket No USA 143) can be booleaned with a fill pattern descπption (e g Style file information as described in Docket USA 143) to be used for the particular region to derive a final bit map representation used for firing the dispensing jets The actual control of the jets may be governed by a subsequently modified bit map which contains data which has been skewed or otherwise modified to allow more efficient data passing to the firing control system These considerations are discussed further in the U S Patent Application based on 3D Systems' Docket Number USA 143 The raster lines making up the grid are then assigned to individual orifices in the manner described earlier Then, a particular orifice is directed to fire or not at firing locations corresponding to desired drop locations or pixel locations depending on how the corresponding cells in the bit map are flagged As discussed above the print head 9 is capable of depositing droplets at many different resolutions In some preferred embodiments of the present invention SDP = SDL = 300 pixels and drops per inch Also in some preferred embodiments, MD L is allowed to take on three different values while MDP remains fixed 1 ) MDL = 300 drops per inch and MDP = 300 pixels per inch, 2) MDL = 600 drops per inch, and MDP = 300 pixels per inch or 3) MDL = 1200 drops per inch and MDP = 300 pixels per inch. When the MDL to MDP ratio is greater than one, the extra drops per pixel are made to occur at intermediate locations (ID overpπnting) between the centers of the pixels With the currently preferred pπnt head and material, the volume per drop is about 80 to 100 picoliters which yields roughly drops having a 2 mil (50 8 μm) diameter With the currently preferred pπnt head, the maximum frequency of finng is about 20 Khz By way of companson, a finng rate of 1200 dpi at 13 ips involves a firing frequency of about 16 Khz, which is within the permissible limit In some preferred embodiments, build styles are defined separately from object data for ease of data manipulation, transfer and memory loading In this regard, as noted above data descπptive of the object is Booleaned (e g intersected) together with information descriptive of a build style, on a pixel by pixel basis to yield a pixel by pixel representation of the deposition pattem at any given location For example, if a completely solid pattem is to be dispensed in two passes (e.g a two step pattem), the object data would first be Booleaned (e g intersected) with a first build style pattern representing the portion of the pixels at which drops are to be deposited (or for ease of terminology we may say 'exposed" in analogy to the selective solidification that is used in photo-based stereolithography) The resultant modified pixel data could thereafter be used to control jet firing Next, the object data would be Booleaned (e g intersected) with the complementary build style pattem to yield modified pixel data for controlling a second finng of the jets In other preferred embodiments, object data and support data can be immediately correlated to build style data upon its derivation In further preferred embodiments, build style information could also include pixel shifting information, pixel sizing information ovβφπnting information, scan direction preferences for depositing on each pixel location, planarization direction and rotational preferences, and the like The build styles descnbed herein enhance system performance by 1 ) enhancing build speed 2) enhancing accuracy of the formed object 3) enhancing surface finish 4) reducing stress in the object and/or distortion of the object or 5) a combination of one or more of these simultaneously A significant problem with Selective Deposition Modeling systems involves ensuring the reliability of mateπal deposition and more particularly of achieving uniform thickness of deposited cross-sections Another problem involves achieving a uniform thickness for all build styles In ink jet systems this reliability problem can take the form inter alia, of misfiπng or non-fiπng jets In a multijet system, further problems exist regarding non-uniformity of jet firing direction non-uniformity of dispensed volume between jets, and to a lesser extent, non-uniformity of dispensed volume from a single jet over time The problem of non-uniformity of cross-section thickness can also result from other phenomena as well As an example, once a droplet leaves a jet there is a time of flight before the droplet encounters the working surface When leaving the jet, the drop is fired with an initial downward velocity component away from the jet but since the jet is moving in the main scanning direction the droplet also has a honzontal velocity component Once the droplet leaves the jet it is subject to vaπous extemal and internal forces including gravity, viscous drag forces and surface tension These initial conditions and forces in turn lead to the conclusion that the droplet may not, and probably will not, land directly on the working surface below the position from which it was fired Instead the droplet will land somewhat away from this theoretical drop point, typically in the direction of travel of the pπnt head In other words the finng location and impact (or drop) location will not have the same XY coordinates but instead will be shifted one from the other The shift in honzontal distance that occurs depends on the above noted factors but also on the distance between the oπfice plate 10 and the vertical position (e g "Z" position) of the working surface at each honzontal location (e g X and/or Y position) As noted above vanatioπs in vertical position can occur for a number of reasons For example, vaπations can result from differences in geometry between different portions of a cross- section (more or less mateπal spreading results in less or more deposition thickness) As another example, vaπations can result from the temporal ordeπng of deposition for a given spatial pattern (previously deposited material on an adjacent pixel site can limit the ability of the material to spread in that direction). As noted previously, the preferred system for embodying this invention utilizes planarization to bring each deposited cross-section to a uniform height wherein the net layer thickness results from the difference in Z-level between the planaπzation levels of two consecutive layers In turn, if it is desired that the planarization step form a completely smooth and uniformly leveled layer, the Z increment between planaπzations must be at or below the minimum deposition/build up thickness for each point on the entire layer If one jet is weakly firing (or not firing), the minimum thickness build up can result in net layer thicknesses much smaller (i e near zero or zero) than desired and therefore much longer build times than desired Several techniques for dealing with these deposition/build up problems are described herein Other preferred embodiments might involve the use of planarization on periodic layers instead of on every layer For example plananzation may be used on every second, third, or other higher order spaced layer Alternatively, determination of which layers or portions of layers to planaπze may be based on object geometry Time of Flight Correction: As noted above, one difficulty in ensuring that the droplets strike the desired locations on the working surface involves the time that the droplets are in flight (i e the time of flight of the droplets) If the times of flight were always the same and if the direction and amount of offset were always the same there would be no time of flight issue since the only effect would be a shift between firing coordinates and deposition coordinates However, when forming three-dimensional objects it is typically desirable to jet material when the head is traveling in both the positive and negative main scanning directions (and may even involve, for example, alternating the definitions of main and secondary scanning directions) This results in a change in offset direction (e g reversal of offset direction) between scans due to relative movement occurring in different directions (e g. opposite direction) This problem can be readily addressed by causing firing signals to occur before the head actually reaches the point directly above the desired deposition site This correction to firing time is known as the "time of flight correction The time of flight may be corrected by utilization of a correction factor applied to scanning in each direction separately or alternatively a single correction factor may be used to bring deposition from one scanning direction into registration with the uncorrected scans made in the other direction The time of flight correction may be implemented in a number of ways One way, for example is by appropnately defining the initial finng location (X position) at the beginning of each raster line, which initial firing location will be used to set the firing locations for all other pixels along the raster line Figures 27a-27e illustrate the relationships between firing location, drop location, and time of flight wherein like elements are referenced with like numerals. Figure 27a illustrates the situation where finng locations 404a and 404b are both coincident with desired drop location 402 (i.e. no time of flight correction factor is used). Element 404a represents the finng location when the head is passing in the positive X-direction, represented by element 406a, and element 404b represents the firing location when the head is passing in the negative X-direction, represented by element 406b Elements 408a and 408b represent the nominal path followed by the droplets after leaving firing locations 404a and 404b, respectively The nominal paths 408a and 408b direct the droplets to actual drop locations 410a and 410b, where the droplets impact the surface and foπn impacted droplets 412a and 412b The cross-over point (i e focal point) for the droplets fired, while scanning in both directions, is depicted with numeral 414 The plane defined by the cross-over points for the entire layer may be called the focal plane Elements 416a and 416b represent the time of flight factor used in terms of an X-displacement between the firing locations and the desired drop location Whether or not the actual drop locations match the desired drop location determines the appropπateness the correction factor In Figure 27a it can be seen that the droplets are moving in diverging directions and that the impacted droplets do not overlap at the working surface resulting in a minimal build up in Z and inaccurate XY placement of material Figure 27b represents the situation where a small time of flight correction factors 416a and 416b are used which result in a focal point located above the desired working surface and in a closer spacing of the impacted droplets 412a and 412b as compared to that depicted in Figure 27a If the time of flight correction were any larger, Z build up would be increased due to the overlap or superposition of impacted droplets 412a and 412b Figure 27c represents a situation where the time of flight correction factors used result in the most accurate placement of impacted droplets 412a and 412b (assuming the thickness of impacted droplet 412a is small compared to the drop distance 418 and that the angle of incidence is not too large) If optimal time of flight correction is based on maximum Z accumulation then Figure 27c depicts the optimal situation Figure 27d represents the situation where the time of flight correction factors 416a and 416b are slightly larger than those used in Figure 27c but still result in Z-accumulatιon based on the supeφosition of both droplets The X-direction placements of the droplets are still reasonably accurate and the focal point 414 of dispensing is somewhat below the desired working surface (and actual working surface) Figure 27e represents the situation where even larger time of flight correction factors are used such that Z-accumulation is reduced to a minimal amount and wnere the focal point is even further below the desired working surface If drag effects and gravitational effects on flight time are ignored, the time of flight correction value (time) would be equal to the distance (length) separating the orifice from the working surface divided by the downward velocity (length/time) at which the droplet is dispensed However, it is believed that drag is an important factor For example, in some preferred embodiments print head scanning speed is approximately 13 inches per second, the distance from the orifice plate to the working surface is approximately 0 020 inches, and the initial vertical firing speeds are believed to be on the order of 200 to 360 inches per second If drag or other frictional forces are ignored, under these itial conditions, a shift between firing locations and drop locations of approximately 0.8 to 1 3 miis would be expected However, under these conditions, in practice shifts in the main scanning direction between the finng location and drop location of approximately 2 mils have been observed The appropπate correction value can be readily determined empiπcally by attempting to deposit droplets at a single X location when scanning in both directions and reiterating the experiment with different correction values until the two drops land at the same point. As noted above, in some preferred embodiments the most appropriate time of flight correction value is the one for which the droplets hit the same position. In terms of the above example, if drag forces are ignored, time of flight correction factors of approximately 60 to 100 μS would be expected. When in practice correction factors of approximately 150 to 200 μS have been found to be more appropriate. In other preferred embodiments the optimal time of flight correction factor is not set at a value which yields the most accurate targeting (i e. the focal point is not at the working surface) but instead is set a value which would yield most accurate targeting some distance below the actual working surface (i.e. the focal point is located below the working surface) These embodiments may be called "off surface targeting" embodiments In this context, most accurate targeting is considered to occur when vertical accumulation rate is the greatest and probably when the X position is most precisely impacted Figure 27d depicts an example of targeting for these off surface targeting embodiments. These off surface targeting embodiments are believed to be particularly useful when building is to occur without the use of additional components for maintaining the desired and actual working surface at the same level (e.g without a plananzer or without additional elements such as a surface level detection device and adjustment mechanisms or schemes) A characteristic of these off surface targeting embodiments is that Z-accumulation is self- correcting or self compensating As long as the Z-increments between deposition of successive layers are within an appropriate range and the deposition pattern allows honzontal spreading of dispensed material instead of only vertical accumulation, excess Z-accumulation on one layer causes a reduction in Z-accumulation on one or more subsequent layers causing the net accumulation to maintain the focal point somewhat below the actual working surface. On the other hand, again as long as Z- increments between deposition of successive layers is within an appropnate range and the deposition pattem allows honzontal spreading of dispensed matenal instead of only vertical accumulation, too little Z-accumulation on one layer causes an increase in Z-accumulation on one or more subsequent layers thereby causing net accumulation to maintain the focal point somewhat below the actual working surface. The preferred Z-increment range is discussed further below This self correcting aspect can be understood by studying and comparing Figures 27c, 27d and 27e. When deposition begins (e.g. at the platform) the time of flight correction factor(s) are chosen such that the focal point is somewhat below the actual working surface as depicted in Figure 27d (i.e. the focal point should be set at an appropπate position such that the situations depicted in Figures 27c and 27e do not occur). If when forming the first layer, too little mateπal is deposited, for the given Z-increment being used, the actual surface will be lower as compared to the repositioned focal plane (but will still be above it as long as the Z-increment was not too large). This results in a more optimally focused deposition when forming the next layer, this in turn results in an increase in deposition thickness as depicted in Figure 27c. If the net Z-accumulation resulting from depositing the second layer is still too low (as compared to the two Z-increments made), then the next layer when being deposited will have an actual surface which closer to the optimal focus plane than the original surface was. This closer approach to optimal positioning results in increased Z-accumulation which will again drive the net accumulated thickness toward that required by the Z-increments On the other hand, if net accumulation from depositing the second layer is greater than that dictated by the two Z- increments, then the actual working surface will be further away from the focal plane and less Z- accumulation, upon forming the next layer, will occur thereby driving the net accumulation toward the amount required by the Z-increments This is the situation depicted in Figure 27e When the focal plane is appropπately below the actual working surface, when the z-increment amount is appropriately selected to approximately match deposition rates, and when objects/supports are being formed in a non-solid manner (not all pixel locations are directly deposited on, the system is stabilized and both supports and objects can be formed with accurate vertical dimensions without the explicit need of a planaπzer Of course a plananzer can still be used if desired For optimal operation of these embodiments it is preferred that the Z-increment should be selected to be between the average amount accumulated per layer during optimal targeting (e g Figure 27c) and the average amount accumulated when no superposition occurs (e g Figure 27e) It is further preferred that layer thickness be significantly less than the distance that separates the plane of optimal focus (e g Figure 27c) from the plane where supeφosition no longer occurs (e g figure 27d) As noted above, in some of these embodiments objects may be formed in such a manner as to allow regions for material to spread horizontally instead of just accumulating vertically, based on the level of targeting optimization, and thereby allowing self correction of Z-accumulation One such embodiment might involve the forming the object as a combination alternating solid layers and checker board layers Other such embodiments might involve the formation of solid outward facing surfaces and checkerboard, offset checkerboard or other open structures in internal object regions Other appropπate building pattems can be determined empirically by building and analyzing test parts In some of these offset surface targeting embodiments, the most preferred initial target surface/focal plane position is selected to be approximately in the middle of the situations depicted in Figures 27c and 27e One way of accomplishing this is to ignore the hypothetical focal points and instead focus on time of flight values The time of flight correction values may be selected so that they are greater than the optimal time of flight correction values (as discussed above) and less than the time of flight correction values which yields immediately adjacent but non-overlapping (i e non- supenmposed) impact zones Most preferably the selected time of flight values would be taken as approximately the average of these two extremes Some offset surface targeting embodiments might be used to simultaneously form different portions of objects and/or supports such that their upper surfaces are intentionally at different heights after formation of any given layer These different height embodiments might benefit from utilization of data manipulation techniques like the SMLC techniques, discussed in previously referenced U S Patent Application No 08/428,951 as well as some of the of the other previously referenced U S Patents and applications In addition to the above noted time of flight issues, other issues aπse that can be corrected using modified time of flight correction factors For example, when using ID oveφπnting techniques to cause more build up, features on scan lines which are scanned in opposite directions will lose alignment since the feature will be extended in one direction on one line and in the other direction on another line. This situation is depicted in Figures 17a and 17b. Figure 17a depicts two points 60 and 100 belonging respectively to scan lines traversed in directions 64 and 104. Regions 62 and 102 depict the extends of deposited mateπal associated with points 60 and 100, respectively. Figure 17b depicts the same points 60 and 100 where jetting occurs using four times overprinting (i.e. four droplet depositions per pixel) Extents of deposition are depicted with numerals 76 and 106 respectively As can be seen, due to the different directions of overprinting, registration between the physical features on the two lines is lost. The above mis-registration can be corrected by an additional time of flight correction factor which can be empirically, or possibly theoretically determined so as to cause realignment of features on different scan lines. Of course this form of correction does not account for any extra length added to object features along the scanning lines. A different foπn of correction that can avoid both problems is proposed which involves recognition that a given pixel is not bounded on its far side, in the scanning direction, by an adjacent pixel that also calls for mateπal deposition Based on this recognition, no overprinting is used on such an unbounded pixel As another alternative, the extra line length might be compensated for by using a form of drop width compensation similar to line width compensation used in photo-based stereolithography and as described in the previously referenced U.S Patent Application Nos 08/475,730 and 08/480,670, but applied only to the points along each scan line representing a transition from deposition to no deposition As an approximate correction these "terminal points" could simply be deleted from the deposition pattem as they will be in the range of Vi to fully covered by the use of ID overpπnting of immediately adjacent pixels. Another variant involves the use of shifted time of flight correction data to implement subpixeling deposition The time of flight correction factors may also be used in variant manners for somewhat opposite purposes to those described above In these embodiments, time of flight correction factors may be used to deposit mateπal at intermediate pixel (i.e. subpixel) locations for implementation of enhanced building techniques These enhanced building techniques might involve formation of down- facing surfaces, formation and placement of supports, enhanced vertical build up of material, enhanced resolution, and the like. In preferred embodiments, enhanced object formation may be achieved in a single pass or multiple pass implementations Droplet Width Compensation: In some situations it may be desirable to modify the object data by performing droplet width compensation (i.e. deposition width compensation). Compensation (by offsetting inward toward solid one or more full pixel widths) can be used to achieve enhanced accuracy if the drop width is at least somewhat greater than the pixel width and/or length. This technique may be used in combination with any of the embodiments described above or any embodiments descnbed herein after As the drop width approaches or exceeds twice the pixel width (and/or length) better and better accuracy can be obtained by a single or multiple pixel offset. Droplet width compensation may be based on techniques like those disclosed in U.S Patent Application Nos. 08/475.730 and 08/480,670. Alternatively they may involve pixel based erosion routines In some embodiments the pixel based erosions might involve multiple passes through a bit map wherein "solid" pixels meeting certain criteria would be converted to "hollow" pixels Some embodiment might involve the following steps wherein each edge of the bit map is 1 ) In a first pass through the bit map all "solid" pixels which are bounded on their right side by a "hollow" pixel are converted to "hollow pixels, 2) In a second pass all "solid" pixels which are bounded on their left side by a "hollow pixel are converted to 'hollow" pixels, 3) In a third pass all "solid" pixels which are bounded on their upper side by a 'hollow pixel are converted to "hollow' pixels, and 4) In a fourth pass all "solid" pixels which are bounded on their lower side by a "hollow' pixel are converted to "hollow" pixels Other embodiments might change the order of steps ( 1 ) to (4) If more than a one pixel erosion is required steps (1) to (4) can be repeated as multiple times until the correct amount of reduction is achieved These embodiments can perform a reasonable droplet width compensation, however, they suffer from the short coming that pixels in solid corner regions (whether an object corner or an object edge that doesn t run parallel to either the X or Y axis) are removed at a faster rate than pixels in which represent boundary regions that are parallel to either the X or Y axis Other embodiments which attempt to address these differentials in erosion rate might involve steps as follows 1 ) In a first pass through the bit map all "solid" pixels which are bounded on their right side by a "hollow pixel and all other sides by "solid" pixels are converted to "hollow' pixels, 2) In a second pass all "solid" pixels which are bounded on their left side by a "hollow" pixel and on all other sides by "solid" pixels are converted to "hollow" pixels, 3) In a third pass all "solid" pixels which are bounded, on at least their upper side, by a 'hollow" pixel are converted to "hollow" pixels and 4) In a fourth pass all "solid" pixels which are bounded, on at least their lower side, by a "hollow' pixel are converted to "hollow" pixels Other embodiments might change the order of steps (1 ) to (4) or the conditions on which conversion will be based If more than a one pixel erosion is required, steps (1 ) to (4) can be repeated as multiple times until the correct amount of reduction is achieved These embodiments do a better job of minimizing excess reduction in comer regions Other embodiments, might involve setting erosion conditions based on whether or not two, three or all four sides of a pixel are bounded by ' hollow" pixels Other embodiments may vary the erosion conditions depending on how many times the bit map has been passed through Other embodiments may use a combination of erosions and Boolean comparisons with original cross-section or other partially compensated bit maps to derive final bit map representations of the pixels to be exposed Numerous other embodiments and algorithms for eroding pixels while emphasizing the reduction or maintenance of certain object features will be apparent to those of skill in the art in view of the teachings herein In situations where X and Y pixels dimensions are significantly different, droplet width compensation may only be necessary along one axis instead of both axes In these situations, embodiments similar to those descnbed above may be used wherein only the a portion of the steps will be performed per erosion It is anticipated that deposition width compensating schemes can also be utilized using subpixel offset amounts in either one or both of the X and Y dimensions Randomization: A technique (method and apparatus) known as randomization can be employed in the build process This technique may be used in combination with any of the embodiments descnbed above or any embodiments described herein after According to this technique, the manner of dispensing material at each location for two consecutive cross-sections is varied This can lead to a more uniform build up of material across a layer (i e lamina) resulting in the ability to potentially use thicker layers, thus improving build time This technique also minimizes the effects from any single jet or plurality of jets that may not be properly finng The varying of deposition can occur in several ways For example variation may occur by 1 ) varying the jet which deposits material onto a given portion of a layer relative to the jet that deposited material on the corresponding portion of the immediately preceding layer, 2) varying the temporal order or spatial order of dispensing onto any given portion of the layer relative to any other portion of the layer and 3) a combination of these, such as varying the mam scanning orientation or direction and/or varying the secondary scanning orientation or direction The varying of deposition from layer to layer can occur in a totally random manner or it can occur in a periodic or planned manner A similar technique has been used in photo-based stereolithography though for a completely different purpose (see Alternate Sequencing in previously referenced U S Patent Application No 08/473,834) Specific embodiments for varying the deposition will now be given The presently preferred randomization technique maintains the orientation of the main and secondary scanning directions but uses a different dispenser (e g jet) to deposit material along corresponding scanning lines between two layers In other words, a first dispenser is used to scan a particular main scanning line on a first layer and a second dispenser may be used to scan that particular main scanning line on a subsequent layer (the one immediately above the particular scanning line on the first layer) In some preferred embodiments, a particular scan line is exposed (i e deposited on), from layer-to-layer, using a different jet until 96 layers have been deposited and each of the 96 jets has deposited on the particular scan line, after which the process repeats These embodiments are examples of "full head" randomization In other preferred embodiments, half head" randomization is preferred Half head randomization can reduce the number of passes that must be made over the any cross-section depending on the object geometry Based on building with the presently preferred 96 jet head, half head randomization involves scanning over any given location with randomized dispensing occurring from either jets 1 to 48 or jets 49 to 96 To explain the full head randomization embodiments in more detail, reference is made to Figures 4a and 6 For a particular layer, orifice 10(1 ) might be used to trace scan lines R(1 )-R(8), orifice 10(2), lines R(9)-R(16), orifice 10(3), lines R(17)-R(25), oπfice 10(4), lines R(26)-R(33), etc On the next layer, however, these assignments are changed such that a given orifice does not trace the same scan line on the next layer For example, the following new assignments might be made onfice 10(1), lines R(257)-R(264), orifice 10(2), lines R(265)-R(272), orifice 10(3), lines R(273)-R(280), etc Another embodiment might involve relatively rotating the partially formed object and/or the pπnt head by some amount (e.g. 30° , 60°, or 90°) between the deposition for two layers so that the main and secondary scanning orientations are changed from their previous orientations. This results in material deposition on a current layer, (i.e. lamina), from any jet, mainly occurnng above material which was deposited by other jets on the previous layer. This is depicted in Figure 8 wherein the scan lines associated with a first layer are depicted by lines R.(1 ), R,(2), R,(3), R,(4) R,(N-3), R,(N-2), R,(N-1 ), R^N) while the scan lines associated with a subsequent layer are depicted by line R 2 (1), R 2 (2), R 2 (3), R 2 (4), . . ,R 2 (N-3), R 2 (N-2), R 2 (N-1), R 2 (N) which are rotated by 90° with respect to the scanning lines of the first layer. The amount of rotation may vary between subsequent layers or it may be a constant amount. The angles may be chosen such that if the rotation is continued for a sufficient number of layers, identical jets will deposit material over identical scan lines where deposition occurred on previous layers Alternatively, the angles may be chosen so that no identical jet to scan line redeposition occurs. Additional embodiments might involve changing the order of progression from one scan line to another (in the secondary scanning direction) This is depicted in Figure 9. where for a first layer the order of deposition of material on main scan tines begins on the upper most scan line. R 3 (1 ) and proceeds to scan lines R 3 (2), R 3 (3) R 3 (N-2), R 3 (N-1), and ends with lower most scan line R 3 (N). The order of progression of scan lines is depicted by arrow R 3P . The deposition of material on scan Iines for a subsequent layer begins on the lower most scan line, R 4 (1 ) and proceeds to scan lines R 4 (2), R 4 (3), ..., R 4 (N-2), R 4 (N-1 ), and ends with upper most scan line R 4 (N) The order of progression of scan Iines on this subsequent layer is in the opposite direction to that of the lines on the first layer and is depicted by arrow R 4P . Additional embodiments are depicted in Figures 10a and 10b, wherein the direction of scanning along corresponding scan lines is reversed between two subsquent layers. Figure 10a depicts the scanning directions for scan lines on a first layer wherein scan lines R 5 (1 ) and R 5 (3) are scanned from left to right and scan line R 5 (2) is scanned from right to left Figure 10b depicts that the scanning directions are reversed on a subsequent layer wherein scan lines R 6 (1), R β (2), and R 6 (3) overlay R 5 (1), R 5 (2), and R 5 (3), respectfully, and wherein scan lines R 6 M) and R 6 (3) are scanned from right to left and scan line R 6 (2) is scanned from left to πght. Many other randomization pattems are possible including combinations of the above descnbed techniques. Depending on the randomization technique chosen, the randomization process may cause an overall increase in layer deposition time since it might result in the need to perform additional main scanning passes. However, this possible disadvantage is believed to be outweighed by the enhancement in uniform layer build up. Additionally, since heat removal is a significant problem when using elevated dispensing temperatures (as used to make the material flowable), these extra passes can be effectively used to allow additional cooling to occur pnor to deposition of a subsequent layer. Drop Location Offsetting: As noted above some building techniques can be enhanced by the use of offset scan lines and/or offsetting of drop locations along scan lines These offsetting techniques could be used in combination with the above noted randomization techniques, though it should be understood that corresponding lines and drop locations on successive layers may be offset from one another These techniques may also be used in combinations with other embodiments disclosed herein before or herein after In some preferred embodiments this offsetting may be up to Vi the line spacing or drop spacing One use of offset pixelling might involve deposition of material on a down-facing portion of a cross-section so as to aid in bridging the gap between adjacent support elements In fact the down- facing region may be cured in multiple passes wherein progressive or alternating offset between successive passes, are used to bridge a wide gap between support elements In these embodiments any non-down-facing portion of the cross-section might be exposed using one or more depositions and offset or non-offset pixels and deposition in any down-facing portion might occur by multiple depositions (or exposures) where pixels zones partly overlap The overall height of deposition, in preferred embodiments might be maαe uniform by trimming to an appropriate level by planarizing In some embodiments offsetting of pixels and therefore drop sites might occur during support structure formation to enhance the formation of arch-like supports bridges, or branching supports (e g like limbs of a tree) In some embodiments, offsetting of pixels might occur during object formation to enhance building of object sections which protrude a limited amount beyond the boundaries of the immediately preceding object lamina Protruding supports and object portions can be formed without the use of offset pixeling but it is believed that offset pixeling can be useful to aid in the formation of such structures wherein less material may slump into regions below the layers levels at which it was dispensed Embodiments may involve the offsetting of pixels on every layer or alternatively may involve the offsetting of pixels only on periodic layers In this last alternative, material is deposited according to the same pixel positions over a number of layers According to this alternative stabilization of overhanging regions may be allowed to better occur by build up of multiple layers above an initial overhang, prior to attempting the formation of a subsequent overhang The offsetting of pixels, e g to build branching supports or tapering outward object structures, results in the formation of structures which branch out over empty space The extent of this branching is limited to some thing less than one droplet width per layer Whether each layer extends beyond the boundary of its immediately preceding layer, or whether multiple layers are built above one another followed by periodic extensions beyond the boundary of an immediately preceding layer, one can define an angle of extension based on the average extension over a number of layers The maximum angle of extension depends, in part on the rate at which the material near and in the extended portion solidifies, which in turn depends on the amount of material dispensed near and in the extended portion The layers can be built at any angle wherein the matenal solidifies quickly enough and is capable of supporting the next layer of material In some preferred embodiments, extension angles close to 30 degrees have been achieved It is believed that extension angles approaching or even exceeding 45 degrees are possible Due to material cool down rates it is preferred that the formation of overhanging solid object portions be achieved in multiple passes In one preferred embodiment, the extension region is deposited on one or more initial passes and the fully supported regions are exposed on one or more subsequent passes This embodiment allows the material in the extension regions to cool and solidify without the added delays that might be associated with heat absorption from material dispensed in the interior regions In another preferred embodiment the interior of the layer is exposed initially and the extension regions are exposed in one or more subsequent passes This embodiment allows time for the material on the interior portions to cool somewhat prior to dispensing the extension regions thereby reducing the risk of the extension material remaining flowable for too long a time For a given set of build parameters, the useable extension angles can be empirically determined by building and examining test parts Offsetting of pixels may be used in combination with multiple passes over a given portion of a layer so as to allow build up of mateπal around a given geometric feature in a prescribed order and offset pattern For example offsetting may occur on one side of a feature such that a Traction of a pixel shift in position occurs away from that siαe of the feature while a different offset might be used so that the same fractional shift might be obtained in the opposite direction on the opposite side of the feature An alternative to offset pixelling is simply to build objects using higher resolution data and associated build patterns or styles that yield the desired drop density which may be less than that provided inherently by the data but which still may yield formation of solid structures or other desired features Scan Line interlacing: Interlacing is another technique that can be used to enhance object formation As with all other embodiments disclosed herein the embodiments of this section are combinable with the those other embodiments disclosed herein As discussed previously, if the head is not oπented at the saber angle, the spacing between the jets is not equal to the desired resolution and thus is not equal to the desired spacing of main scanning or raster lines As such, by its nature, a form of mam scan line interlacing must be used if it is truly desired to deposit material along all main scan lines However, additional interlacing may be done for a number of reasons (e g to enhance layer cooling and/or mateπal build up) A vaπety of scan line interlacing patte s can be used, whether or not the print head is oriented at the saber angle, whether or not the preferred raster scanning technique is used, whether or not a vector scanning technique is used, whether or not some other scanning technique or combination technique is being used In one preferred embodiment, as previously described, the head is oriented perpendicular to the main scanning direction and a resolution of 300 scan lines per inch is used In this configuration successive jets are spaced 8/300 of an inch apart The head is made to perform 8 main scans, the first 7 of which are followed by a secondary scan of width equal to the spacing between the raster lines (raster width), and the eighth of which is followed by a secondary scan of width equal to the effective head width plus the raster width. Repetitions of the above scanning pattern are made until the width incremented in the secondary scanning direction has equaled or exceeded the width of the building region. Alternative embodiments could limit the X range of main scanning to that which is sufficient to effectively cover the working region required by the object, by the particular object cross-section being scanned, by each segment of the object length required to make the 8 closely spaced main scans, or by other schemes which lead to a reduction in scanning time Similarly, the positioning along the secondary scanning axis could likewise be limited to the width and position of the object, cross-section being scanned, particular portion of a cross-section being scanned, or the like. In preferred embodiments, the use of randomization may increase the amount of indexing needing to be performed so that the appropriate jets may trace the appropriate main scan lines Other embodiments may limit the main scanning to paths which actually include active drop locations As a first preferred alternative interlacing technique non-adjacent scan lines would be left unexposed after at least a first pass wnereafter on one or more subsequent passes the intermediate lines would be exposed In other preferred embodiments, it is desired that intermediate raster lines be drawn prior to depositing material on either adjacent raster line or after depositing material on both adjacent lines Examples of this type of embodiment are depicted in Figures 11a, 11 , and 22a - 22d Figures 11a and 11b depict the situation wherein every other line is skipped on a first pass. Figure 11a depicts four scanning lines wherein two lines are to be exposed (i e. the drop locations to be used) on a first pass Figure 11 b depicts the same four scanning lines wherein the other two lines are to be exposed (i.e the drop locations to be used) on a second pass. Further examples of interlaced patterns are shown in Figures 22a-22d. In these figures two headed arrow 30 represents the main scanning direction, spacing d r represents the spacing between successive raster lines, and for claπty, the beginning points and end points of the lines are shown offset although in practice the lines would have the same beginning and ending points Figure 22a depicts a series of raster lines to be scanned in the main scanning direction Figure 22b depicts first raster lines 32 to be exposed on a first pass and second raster lines 34 to be formed on a second pass according to the example of Figures 11a and 11b. Figure 22c depicts raster lines 32, 34, 36 and 38 to be exposed on first, second, third and fourth pass, respectively Figure 22d depicts raster lines 32, 34, 36, 38, 40, and 42 to be exposed on first, second third, fourth, fifth and sixth pass, respectively In the example of Figure 22d other raster line scanning orders could also be used while still ensuring that when intermediate lines are deposited they are either not bounded on either side or that they are bounded on both sides by previously deposited adjacent raster lines For example, other useful scanning orders might be 32, 34, 38, 36, 40 and 42, 32, 36, 34, 40, 38 and 42, or the like In one preferred system, to fully implement these embodiments in a generalized manner using a minimum number of passes, an odd number of raster lines would need to exist between the line scanned by one of the jets (e.g a first jet), and the line scanned by an adjacent jet (e.g. a second jet). In other words, the number of d. spacing between successive jets would have to be even; thereby requiring that two adjacent jets must be positioned so as to scan raster lines M and M+N where M and N are integers and N is even In the case where the spacing between the jets is not appropnate (e.g not even), it is always possible to scan only appropriate raster lines (e g those associated with every other jet) in a first pass and then to expose the remaining scan lines in one or more subsequent passes As the width of deposition may be significantly wider than the raster line spacing, other preferred embodiments might not be based on the skipping of every other scan line on a first pass, but instead be based on the selection of scan lines for deposition (i e exposing) on the first pass such that the lines of deposited material do not directly contact each other and then filling in any skipped raster lines on one or more subsequent exposures This first alternative interlacing technique can be fully or approximately implemented even when the adjacent jets are inappropriately positioned for the desired scan line resolution (i e the jet positions and scan line resolution are such that an even number of raster lines exist between the line scanned by one of the jets and the line scanned by an adjacent jet) This may be accomplished in at least three ways 1 ) each jet is used to scan every other raster line between its initial position and the position of the line initially formed by the adjacent jet except at least two adjacent raster lines to be scanned by each jet will be left unexposed until at least a second pass when the remaining raster lines will be exposed, 2) each jet is used to scan every other raster line until it also scans the raster line adjacent to the first line scanned by the adjacent jet whereafter the remaining unexposed lines will be selectively exposed in a second pass, and/or 3) only every other jet is used in the scanning process thereby ensuring that an odd number of raster lines exist between any two adjacent jets In these embodiments, as well as the previous embodiments, it is preferable to expose alternating lines for the whole layer pπor to beginning a second pass to expose the intermediate lines, however it is possible to complete the exposure of all scan tines between the starting points of some or all of the adjacent jets pπor to making even a first pass over other portions of the layer Numerous other interlacing embodiments will apparent to those of ordinary skill in the art who study this disclosure For example, interlacing with higher numbers of passes can be used or interlacing wherein some contact occurs between lines exposed on a first pass Of course any combination of interlacing with the previously descnbed randomization techniques could also be used Further exposure of a subsequent layer may change the order of scanning the various sets of lines and/or the scanning directions of the lines themselves (e g reverse the order of scanning of the first, second, and higher order sets) Further embodiments might involve the completion of interlacing exposures for a first layer while exposing regions dunng formation of one or more subsequent layers Drop Location Interlacing: As with scan line interlacing, object formation may utilize drop location interlacing along individual scan lines. In this case, each scan line would be exposed by at least two passes wherein a first pass would expose a number of drop locations and, whereafter, on one or more subsequent passes, the remaining drop locations would be exposed As a two step (i e pass) example, on a first pass every other drop site would be exposed while on a second pass the intermediate drop sites would be exposed. This situation is depicted in Figures 12a and 12b Figure 12a depicts four scanning lines each with 9 drop locations wherein every other drop location is to be exposed on a first pass while Figure 12b depicts the same lines and locations but instead depicts that only complementary drop locations are to be exposed on a second pass. As a second two step example every third site may be exposed on a first pass while on a second pass both the intermediate sites, there between, would be exposed As a three step example, a first pass might expose every fifth site beginning with the first site, then on a second pass every fifth site would be exposed beginning with the third site, and finally on a third pass every other site would be exposed beginning with the second As with all other embodiments disclosed herein, the embodiments of this section are combinable with the other embodiments disclosed herein In these interlacing techniques, successive scan lines may be exposed using different or shifted interlacing patterns so that two dimensional interlacing patterns may be developed (offset pixelling could also be used) For example a two step interlacing pattern may be used on each scan line wherein the starting points on successive lines are shifted by one pixel such that a checkerboard first pass pattern is formed Figures 13a and 13b illustrate this example Figure 13a depicts the first pass checkerboard pattern while Figure 13b depicts the complementary checkboard pattern that is exposed on a second pass As with scan line interlacing, drop location interlacing may complete all passes over single lines prior to exposing subsequent lines though it is preferred that all lines be exposed with each pass pπor to initiating subsequent passes over partially exposed lines Furthermore, completion of all passes over portions of single lines may be achieved prior to initiating exposure over remaining portions of those lines. A third interlacing technique involves feature sensitive interlacing In this technique the order in which a given drop site is exposed depends on the geometry of the immediate cross-section alone or on multiple cross-section geometries Feature sensitive interlacing may involve one or both of scan line interlacing and drop location interlacing For example, in a single layer embodiment one may determine the boundary regions of the cross-sections and ensure that the boundary zones are exposed on a first pass Some interior portions of the cross-section might also be exposed on the first pass or alternatively exposure of all interior portions may be delayed until one or more subsequent passes are made. For example, the inteπor portions may be exposed using a checkboard interlacing pattem on a first pass in combination with all boundary regions also being exposed on the first pass Then on a second pass the remaining interior portions would be exposed It is also possible that a wide boundary width could be defined for exposure on a first pass so that more than a one drop site width border may be placed around the cross-section prior to performing subsequent passes This wide boundary region might be implemented using erosion routines such as those described above in association with Droplet Width Compensation As an additional alternative, one may focus on ensuπng that only one of scan line boundary sites or drop location boundary sites (boundaπes along lines in the secondary scanning direction) are exposed on the first pass As a further alternative internal regions may be exposed in whole or in part prior to dispensing mateπal in boundary regions It is beiieved that dispensing boundary regions first might lead to improved build-up in the vertical direction and that exposing boundary regions last might lead to improved honzontal accuracy of the object An even further alternative might involve the dispensing of a near boundary region initially, followed by the dispensing of deeper internal regions of the cross-section and finally followed by dispensing of the outer cross-sectional boundary itself Examples of a multi-cross-sectional feature sensitive interlacing technique might involve initially exposing those locations which form part of the present cross-section but which were boundary or solid internal object regions on the previous cross-section The boundary and solid internal regions on the previous cross-section might include boundary regions and solid internal regions of support structures as well as object structures In using this embodiment deposition in at least critical (i e important) down-facing object regions does not occur on the first pass unless those down-facing regions are actually supported by a structure of some nature (e g a support column directly below) In one or more subsequent exposures material is dispensed to form unsupported down-facing features Since the deposition width is typically wider than the pixel width, it is more likely that a droplet which is dispensed to land at a pixel location adjacent to previously dispensed mateπal on that cross-section, the droplet will strike and hopefully adhere to the neighboring deposited material as opposed to continue falling downward to a cross-section below that for which is was intended Furthermore, since in preferred embodiments support structures are typically no more than one pixel in separation when exposure of unsupported down-facing regions occur the dispensed mateπal will more likely be wedged between material already dispensed on the present layer as opposed to being wedged between material dispensed on a previous layer However, since droplet diameter is typically less than deposition diameter (i e impacted droplet diameter) and since it may be smaller than pixel width, mateπal deposited at an adjacent pixel location may not sufficiently extend into the path of a falling droplet so as to ensure a collision and stopping of the particle In another preferred embodiment, the drop locations would be shifted by a fraction of a pixel width (preferably approximately Vi a pixel width) along the main and/or secondary scanning directions (preferably both) when dispensing unsupported down-facing regions and preferably adjacent regions such that a droplet is more likely to be at least partially supported by previously dispensed matenal than if droplets were deposited in perfect alignment. It is preferred that droplets over partially unsupported regions be dispensed in a subsequent pass from those dispensed over fully supported regions However, it is possible to rely solely on the overlap with the previous cross-section (and not any additional benefit associated with adhesion to material previously dispensed on the given cross- section) in ensuπng reasonable vertical placement of the droplets in at partially unsupported regions 'n this embodiment at least the support regions (e.g. columns) on the current layer would not be shifted. This ensures that registration from layer to layer occurs. It is further preferred that wide gaps be closed by progressively working deposition locations inward (i.e. multistage) from supported sides of the gap using multiple passes over the cross-section, wherein each pass is partially offset from the immediately preceding pass to ensure adequate overlap of droplets so as to limit any mateπal placement beyond the required vertical level Further, in one preferred embodiment Simultaneous Multiple Layer Curing Techniques, as described in U S Patent Application No 08/428,951, are used in order to offset cntical down-facing data up one or more layers so that upon deposition material forming the down-facing layer will be located at the correct level An example of this multistage horizontally and vertically offset embodiment using a Vi pixel honzontal offset and 1 layer thickness vertical offset is shown in Figures 23a-23h Figure 23a depicts a side view of an object 120 to be formed Figure 23b depicts object 120 as it would normally be divided into layers 122, 124 126 128 and 130 Figure 23c depicts object 120 as it is to be divided into layers 122, 124 126, 128' and 130' Layer 128' is different from 128 in that the down-facing portion of the layer has been removed as it is anticipated that it will be created during deposition of the mateπal on the next layer using a series of successively offset exposures Layer 130' is similar to layer 130 except that a different deposition pattern might be used in its formation Figure 23d again depicts layers 122, 124 126 and 128' but in addition depicts deposition locations, or pixel positions, 132 - 137 at which material may be deposited during formation of layer 130' Figure 23e is similar to Figure 23d except that instead of showing drop locations 132-137 drop locations 140-146 are shown As can be seen from the relative positions of the drop locations, locations 132 - 137 and 140 - 146 are offset from each other by Vi a pixel width Figure 23f depicts the deposition pattem formed from a first pass of the print head in forming layer 130' Droplets 150, 151 , 152, and 153 are deposited at drop locations 141 , 145 142, and 144 respectively It can be seen that droplets 152 and 153 were only partially supported by layer 128' and that as a result it is assumed they will partially extend (as depicted) into the region originally belonging to layer 128 Figure 23g depicts the deposition pattem from the first pass in forming layer 130' as well as additional material deposited in a second pass Region 160 and 162 were deposited on the first pass and were represented in Figure 23f as regions 150, 152, 151 and 153 The deposition on the second pass occurs according to the pixel arrangement depicted in Figure 23d Droplets 155 and 156 are deposited at drop locations 132 and 137 In practice, the dispensing of droplets 155 and 156 would initially result in excess material being applied over a portion of regions 160 and 162 but this excess would be trimmed off during the planarization process Droplets 157 and 158 are deposited at drop locations 134 and 135 but since these locations are not fully bounded from below by material previously deposited, it is assumed that a portion of the material dispensed will extend downward into the region oπginally part of layer 128 The offset dispensing of droplets 152, 153, 157 and 158 results in the formation of the down-facing portion of layer 128 which was removed from layer 128' In a third and final pass, droplet 164 is deposited onto drop location 143 to complete formation of layer 130' In other preferred embodiments various aspects to the above example could be changed For example, the extension of material into lower layer regions (assumed to occur when droplets or drop locations are only partially supported) could take on values other than the 1 layer thickness extension descnbed The extension may be less than 1 layer thickness or at least different from an integral number of layer thickness Maybe the extension would be an integral number of layer thicknesses (e g 2 to 5 layer thickness or more) In such a case, for most accurate formation, it would be desired to have the initial object representation transformed into a modified representation, as descnbed in U.S. Patent Application No 08/428,951, (either before or after generation of cross-sectional data) so that when material is dispensed according to the modified representation, the bottom of the down- facing feature is properly located Other variations might use geometry based deposition, in multiple passes, along with different offset values such as 1/4 of a pixel (so that 3/4 of the drop zone would be unsupported) or 3/4 of a pixel (such that only 1/4 of the drop location would be unsupported) These different offset amounts might lead to more control over the amounts of extension into previous layer regions Other variations might use different deposition orders, different amounts of over printing, or even quantities of deposition per droplet Still other variations might not use offset pixeling but instead would use higher resolution pixels, possibly in combination with deposition patterns yielding the right droplet density An additional interlacing technique combines 1 ) feature sensitivity, and 2) selective direction of scanning when exposing object features In this emboαiment, cross-sectional geometry (e g cross-sectional boundary information) from a current layer and possibly cross-sectional geometry (e g cross-sectional boundary information) from the immediately preceeding layer would be used to determine what the direction of scanning should be when exposing different regions of the cross- section For example, when exposing the left most portion of a solid region of a cross-section it may be advantageous to be scanning the head (i e the jet used for exposing the line to be formed) from left to πght if it is desired that the droplet not bπdge or not partially bπdge any small gaps On the other hand, if it is desired that some bridging occur it may be advantagous to ensure that scanning is in the opposite direction Similarly, when exposing the right most portion of a solid region of the cross- section it may be advantageous to be scanning from right to left (for no bridging) or from left to right (for bπdging) By controlling the scanning direction when depositing boundary regions it can be ensured that horizontal momentum of the droplets either do not contribute to bridging gaps or enhance the bridging of gaps An example of the non-bridging technique is illustrated in Figures 24a-24d Figures 24a-d depict side views of two columns as being formed and as cut in an XZ plane The Z-direction is perpendicular to the planes of the cross-sections and the X-direction is the main scanning direction Reference numeral 108 indicates the cross-section being formed and reference numerals 100, 102, 104, and 106 refer to previously formed cross-sections Figure 24a depicts cross-section 108 with a broken line as no material deposition has taken place Figure 24b indicates that the scanning direction 110 is to the πght and that droplets 112 are deposited on the left most side of each column on a first pass Figure 24c indicates that the scanning direction 124 is to the left and that droplets 114 are deposited on the right most side of each column in a second pass Figure 24d indicates that scanning can occur in either direction 126 and that droplets 116, 118, 120, and 122 are deposited to complete the formation of the cross-section in a third pass As opposed to the illustrated three pass embodiment, a two pass embodiment could be used wherein droplets 116, 118, 120, and 122 may have been deposited to their respective locations dunng one or both of the first or second passes when droplets 112 and 114 were deposited It is anticipated that the object could be relatively reorientated (e g one or more rotations about the vertical axis) with respect to the relative scanning direction of the pnnt head (i e jets) so that the edges of any desired cross-sectional features can be exposed while relatively moving the pπnt head in a desired direction to enhance or decrease the probability of bπdging small gaps As noted above if the orifice plate to working surface distance is too small droplets will have an elongated shape (i e large aspect ratio) as they stnke the working surface In the case of building with elongated droplets it is anticipated that the above indicated scanning directions for depositing on edges of solid features might yield opposite results from those indicated above Other interlacing techniques might involve bidirectional printing of adjacent raster lines or non-adjacent raster lines The above described building techniques can be applied to the formation of solid objects or in combination with other techniques to the formation of partially hollow or semi-solid objects In an original design of an object portions of the object are supposed to be solid (i e be formed of solidified material) and portions are supposed to be hollow (i e empty regions) in actuality these intended hollow (or void) regions that are not supposed to be part of the object since by definition wherever there is object there is supposed to be material In the context of the present invention a non-solid hollow or semi-solid object is an object built or to be built according to the teachings of some preferred embodiments wherein a portion of what should be solid object has been removed A typical example of this might be the hollowing out partial hollowing out or the honeycombing of what was originally a solid structure of the object These originally solid structures are sometimes referred to as object walls regardless of their spatial orientation Some preferred build styles form completely solid objects, while other build styles form solid surface regions of the objects but hollowed out or partially hollowed out inteπor regions For example, the interior portions of an object might be formed in a checkboard cross-hatched hexagonal tiled, or honeycombed manner (these and other build styles useful herein, as implemented in photo-based stereolithography are descnbed in the above referenced patents and applications) The above non-solid deposition patterns can be considered internal object support structures As such the other support structures described herein can also be used as internal object support structures Such non-solid objects would be lighter in weight than their solid counterparts, they would use less material they might even be formed more quickly depending on the details of the specific building parameters and they might be formed with less risk of encountering heat dissipation problems since much less heated material is deposited during their formation These objects might be useful as investment casting patterns due to the decrease in the possibility of cracking molds Temperature Control: Additional object formation embodiments involve the formation of the object wherein the partially formed object is maintained within a desired temperature range as it is being formed or is at least maintained such that the differential in temperature across the part (or the gradient of temperature difference) is small If duπng object formation, the different portions of the object are allowed to be at different temperatures the object will undergo a differential amount of shrinkage as it is cooled to room temperature or as it is brought to its use temperature (the temperature at which it will be put to use). This differential in shrinkage could lead to the development of stresses within the object and associated distortions or even fractures of the object. It is preferred that temperature differential remain within a range which is effective for maintaining object distortion within a reasonable limit. The temperature differential across the object is preferably maintained within a range of 20° C, more preferably within a range of 10° C and even more preferably within a range of 5° C and most preferably within a range of 3° C. In any event, the desired temperature can be estimated by taking into consideration the coefficient of thermal expansion of the material and the differential in shrinkage (or expansion) that would occur upon cooling (or heating) the formed object to a uniform temperature. If the differential in shrinkage results in an error outside a desired tolerance range, the above mentioned ranges of temperatures can be adjusted. In the formation of objects, the initial object data can be scaled to take into account dimensional changes in the object that will occur as the object is cooled from its jetting temperature ( about 130° C in the preferred embodiment) to its solidification temperature (about 50° C - 80° C with a peak DSC energy transfer temperature of about 56 ° C) to its build temperature (about 40° C - 45° C) and finally to its use temperature (e.g. room temperature - about 25° C). This scaling factor could be used to expand the initial object design by an appropriate thermal shrinkage compensation factor such that it would be appropriately sized at its use temperature. It is further anticipated that one or more geometry dependent or at least axes dependent shrinkage factors could be used to at least partially compensate critical regions of the object for expected vaπations in object temperature during build up. The temperature of the previously formed lamina and the cooling rate of the lamina being formed have been found to be important parameters for the formation of objects with reduced distortion and in particular with reduced curl distortion. Presently preferred materials undergo about 15% shrinkage when cooled from their solidification temperature to room temperature. This shrinkage provides a tremendous motivation force for causing curl distortion, build up of intemal stresses, and associated post processing distortions (these distortions are described with regard to photo-based stereolithography in the above referenced patents and applications wherein many of the building techniques described therein can be effectively utilized in the practice of SDM and TSL in view of the teaching found in the instant application ). It has been found that if the object build temperature and in particular if the temperature of the last formed layer is maintained at a temperature above room temperature during the build process, curl distortion will be reduced. It is preferred that the temperature of the entire partially formed object be maintained above room temperature and, more particularly, that its temperature remain within a tight tolerance band due to the differential shrinkage considerations discussed above. For effective object formation, it is apparent that the build temperature of the partially formed object must be maintained below the melting point of the material. Additionally the build temperature must be maintained below a temperature that allows the solidified material to have sufficient shear and compressional strength and even tensional strength (especially if sideways or upside down object formation embodiments are used) to allow the object to be accurately formed while experiencing the typical forces associated with the build process (e g inertial forces associated with accelerations the object will expenence, drag forces or vacuum forces associated with the planaπzer and pπnt head contacting or passing close to the object, air pressure forces associated with any air flow used to cool the object, and gravitational forces on the object due to its own weight) Some of these forces are dependent on the mass of the object and increase with depth into the part Thus, a slight negative temperature gradient from higher to lower layers (i e decrease in temperature from most recently formed layers to earliest formed layers) can supply increasing strength in needed regions while simultaneously allowing the latest formed layer or layers to be at a high enough temperature to result in minimal curl and other distortions One might use a simple gravitational force calculation summed with an inertial force calculation for one or more positions in the part (based on the mass of the part and the Y-direction acceleration it experiences) as an approximation of the minimum shear strength needed from the solidified material This in combination with an empirical determination of the variation of material shear strength with temperature can be used to estimate the approximate upper build temperature limit for any position in the object Of course it is preferred that additional considerations be taken into account especially near the latest formed lamina of the object since dynamic thermal effects occur at the interface of the partially formed object and the material being dispensed that involve remelting phenomena and heat capacity phenomena which are dependent on object geometry parameters, temperature differentials, and cooling techniques Thus, the actual overall maximum build temperature will probably be lower than the above estimated amount On the other hand as noted above, curl and other distortions can be significantly reduced by building at elevated temperatures wherein the higher the temperature the less the distortion It is postulated that this reduction in distortion results from a combination of the material's enhanced ability to flow at elevated temperatures and its lower ability to support shear loads which allow some material redistπbution to occur thereby reducing stress which causes distortion It is further postulated that working near, at, or preferably above any solid state phase change temperatures (e g crystallization temperature or glass transition temperature) will result in the quickest and potentially most significant reductions in stress and distortion Since these phase changes typically occur over a broad range, various levels of benefit are postulated to occur depending on where the working temperature is in within these ranges and the process time allowed Melting temperatures and/or solidification temperatures and solid state transition temperatures can be determined using Differential Scanning Calonmetry (DSC) techniques which in turn can be utilized in determining appropriate build temperature ranges Additionally, appropriate build temperature ranges can be determined empincally It has been determined that some benefit can be gained by working at any temperature above room temperature and it is anticipated that the closer one moves to the melting temperature and/or solidification temperature the more the benefit Thus, the working temperature range might be set as a percentage of the distance along the temperature differential between room temperature and melting or solidification temperature or room temperature and the temperature of estimated minimum shear strength Alternatively, the working temperature may be selected to be a temperature for which the material has a certain percentage of its room temperature shear strength For example it might be desired to set the working (build) temperature such that the shear strength is 75%, 50%, 25% or even 10% of its maximum room temperature value. Surface Finish Enhancement: Additional building embodiments useful for enhancing object surface finish involve taking advantage of the aesthetically pleasing, up-facing surfaces which result from the practice of preferred SDM techniques. In these embodiments the number of effective up-facing surfaces (e.g. the overall area) is increased while the number of effective down-facing surfaces is reduced from that defined by the oπginal object design. This involves splitting the object into two or more pieces and changing the orientation of the separated pieces such that as many critical surfaces as possible are made to be up- facing surfaces, vertical surfaces or combined up-facing/vertical surfaces whereas no truly extemal surfaces or only less cntical surfaces remain as down-facing surfaces. These separate object components are then built independently of one another, each with the proper orientation. Then supports are removed and the resultant components combined by gluing, or the like, such that a complete oDject is formed primarily from up-facing and vertical surface regions. If rough surfaces are desired instead of smooth surfaces, the above technique can be used to ensure that critical surfaces are formed as down-facing surfaces As an alternative the up-facing surfaces which are to be roughened can simply be formed with supports extending therefrom. An example of this building technique is illustrated in Figures 25a-e Figure 25a depicts the configuration of an object 60 to be formed using SDM (i e. the desired object design). If the object is formed directly from this design, the object will be formed with both up-facing features or surfaces (50, 52, and 54) and down-features or surfaces (56 and 58). As discussed previously, the formation of down-facing features requires the pπor formation of a support structure which acts as a working surface onto which mateπal forming the down-facing features is dispensed. After object formation and removal of the supports, it has been found that the down-facing surfaces are left with a rough and irregular surface finish. If it is desired that the down-facing surface be smooth, the object must undergo additional post-processing which may require detailed sanding or filling. Figure 25b depicts the first step in the practice of the above technique. This first step involves splitting the oπginal or desired object design into two or more components. The splitting is performed so that alt cntical features of the object can be formed as either vertical surfaces or up-facing surfaces (preferably as up-facing surfaces and more preferably as up-facing surfaces which do not have down- facing surfaces above them so that supports will not be formed which start from and mar the up-facing surfaces). Additional details about support formation and issues associated therewith will be descnbed further hereinafter. In the present example, all surfaces 50, 52, 54, 56 and 58 are considered to be critical and should be formed as up-facing surfaces. Figure 25b depicts object 60 being split into two portions 62 and 64. Portion 62 includes oπginal outward facing features 50, 52, and 54 and new or temporary outward facing features 72 and 74. Portion 64 includes oπginal or desired outward facing features 56 and 58 and new or temporary outward facing features 72' and 74' Figure 25c depicts the preferred orientation (πghtside up) of portion 62 duπng formation so that surfaces 50, 52 and 54 are formed as up-facing features Figure 25d depicts the preferred oπentation (upside down) of portion 64 during formation so that surfaces 56 and 58 are formed as up- facing features After formation of each portion 62 and 64 the supports are removed and temporary pairs of surfaces 72 & 72', and 74 & 74' are prepared for mating Figure 25e depicts the joining of portions 62 and 64 to form object 60 wherein all critical outward facing portions (i e original surfaces 50 52, 54 56 and 58) have good surface finish Additional Build Styles: Other building styles might include one or more of the following 1 ) the use of higher resolution dispensing in the scanning directions, 2) the use of a higher drop density per unit area in forming down-facing skin surfaces than in forming interior regions of the object, 3) the use of down-facing skin regions which extend at least N layers (e g , 5 to 10) above down-facing surfaces, 4) the use of a higher drop density per unit area when forming up-facing skin surfaces than in forming interior regions of the object 5) use of up-facing skin regions which extend for at least N layers (e g , 5 to 10) below an up-facing surface, 6) the use of higher drop density per unit area when forming boundary regions of an object than when forming interior regions boundary regions which extend at least L drop widths (e g , 2 to 4) into the interior of an object and 7) forming interior regions of the object through raster scanning and boundary regions through vector scanning Support Styles: The next portion of the application is primarily directed to support formation It should be appreciated however, that as supports are formed from deposited mateπal, all the aforementioned building techniques are applicable to the support building process Moreover, as will be appreciated, all aspects of the support building process are applicable to object building as well Support structures must serve several needs which may be opposing 1) they preferably form a good working surface on which to build object lamina and even successive support lamina, 2) they are preferably easily removable from the down-facing surfaces they support, 3) if they start from an up-facing surface of the object, they are preferably easily removable therefrom, 4) when removed, the suports preferably cause only minimal damage to down-facing and up-facing surfaces and preferably have at least a tolerable to good surface finish on those surfaces, 5) they preferably build up at a reasonable rate per cross-section in the vertical direction (e g Z-direction), 6) they are preferably formed using a minimal number of passes per layer, and 7) their formation is preferably reliable A numDer of different support styles have been developed or proposed which achieve different balances between these needs To optimize building speed, vertical accumulation is important and, as such, it is desirable to have supports build up at approximately the same rate as the object In particular, it is preferred that the vertical accumulation of supports (e g , from a single pass per layer) be at least as great as a desired layer thickness set by the use of the plananzer The closer the support accumulation is to that of object accumulation, the thicker the useable layer and the less material that will be removed dunng planaπzation which thereby increases the efficiency of the building process For a given mateπal and apparatus, the vertical build up of material from different support and build styles can be empirically determined, as described previously, by building test parts for each deposition style or pattem using different layer thicknesses (planarization levels) and thereafter measuπng the parts to determine when the build up of material lagged behind the anticipated thickness as dictated by the number of layers deposited and the expected layer thickness From this information one can either set the layer thickness (planarization level) to an appropriate amount for a desired combination of build and support styles or one can set the required support and build styles necessary to achieve the desired layer thickness Some preferred support style embodiments emphasize speed of formation, maintain easy removal, but leave rough surface finish in regions where supports have been removed These support style involves the formation of solid columns which are separated by small gaps In particular, in a preferred system, data is supplied at 300 pixels per inch in both the X and Y directions and the object and supports are formed using four times ID oveφπntmg in the X direction (main scanning direction) Each layer of supports includes three-by-three pixels zones where support material is to be dispensed with the columns separated by two pixels zones of no pixel defined deposition along the main scanning direction (X-direction) and one pixel zone of no pixel defined deposition in the secondary scanning direction (Y-direction) The data situation defining these pixel zones is depicted in Figure 15a The "X's" in the figure depict pixels which contain droplet data while the "O's" in the figure depict pixels which contain "no droplet" data Squares 50 have been inscribed around the "X" zones so as to highlight the shape of the deposition zones However, due to the ID overprinting in the X-direction, the two pixel gaps are actually narrowed considerably (by almost one pixel width) when actual deposition occurs Thus, the actual resulting pattern of deposition more closely approximates 4 by 3 pixel width (12-14 mils by 9-10 mils) columns, though with rounded comers, which are separated by 1 pixel width gap in both X and Y (3 3 mils) This situation is approximately depicted in Figure 18 In the practice of building objects it has been found that supports of the above configuration accumulate at approximately the same rate as the object and thus a single pass of the head over each drop site can be used for forming both supports and object on each layer It has also been found that the above support structure is easily separable from the object but that a poor down-facing surface finish results Therefore, in teπns of building speed, the above style is preferred, but in terms of surface finish, significant room for improvement remains A variant involves using multiple passes of the dispensing head to form a support portion of a cross-section Another alternative involves periodically dispensing an extra support cross-section in order to equalize vertical mateπal accumulation between supports and the object Another vanant involves allowing support formation to lag behind object formation by one or more layers to eliminate or minimize planarization problems that can occur in the case where fragile supports are being built The problem is that the plananzer can cause these supports to distort if support portions of a cross-section are dispensed during the same pass or passes as the corresponding object portion of the cross-section By allowing a lag of one or more layers to occur, excessive contact between the supports and the plananzer can be avoided, and it is anticipated that the resultant distortion of the supports will be minimized Other column-like support structures are possible including columns of different dimensions or shapes For example, data formatting and oveφπntmg techniques could be combined to produce physical columns of approximately a 3 by 3 pixel size (9-10 mils by 9-10 mils), 2-by-3 or 3-by-2 pixel size (these may result in less vertical accumulation), 2-by-2 (6-7 mils by 6-7mιls) pixel size ( probable loss in vertical accumulation rate), 4 by 4 (12-14 mils by 12-14 mils) pixel size (may be more difficult to remove and cause further damage to object surfaces), or even larger sizes Other cross-sectionally shaped columns may also be used These might include more circularly shaped structures (e g octogonal or hexagonal), cross-like structures, structures with different length to width aspect ratios, or combinations of structures that can be intermixed Other alternatives might include offsetting alternate support columns in one or both of the main and subscanning directions For example, every other support column could be offset in the secondary scanning direction by Vi the separation between columns This is depicted in Figure 19 Wider spacing of support columns is possible, particularly if some technique, such as arch or branching supports are used to narrow the gap between the support columns prior to encountering a down-facing surface of the object Two examples of arch-like supports are depicted in Figures 21a and 21b wherein different amounts of pixel offsetting (or at least drop placement control) are used) Branching Supports: As described at several locations herein above, some preferred embodiments utilize supports that may be descnbed as branching supports The arch-type supports discussed above are an example of a type of branch support Branching or branch-type supports are support structures that are built such that portions of some lamina extend outward in a cantilever manner from solidified regions on the immediately preceding lamina These outward extensions may be based on identical (i e fixed) pixel positions from layer to layer Alternatively these outward extensions may be based on fractional pixel width shifts in pixel positions between some or all layers Further alternatives may be based on changing pixel patterns between some or all layers Some embodiments of branching supports produce more individual support structures at a surface to be supported than the number of support structures from which the branching supports onginated at a lower layer In addition to the vaπous embodiments disclosed previously (which in essence can be considered branching supports), Figure 28a, 28b, 29a-e, 30a-m, 31 a-c, 32a-d depict additional examples of preferred branching support structures Figure 28a depicts a side view of column supports 504, 506 & 508 starting at surface 500 and working up toward surface 502 These column supports are connected one to another by branching elements 510, 512, 514 and 516 Figure 28b depicts a side view of an embodiment of branching type supports that work up from surface 500 toward surface 502 The supports are shown to branch every two layers In this two dimensional view, some branching appears to occur in a two path fork-like pattern while other branches simply branch out along a single path. The same support structure depicted in Figure 28b is looked at from a different view in Figures 31 a-c and 32a-d. Other preferred branching pattems are illustrated in the example of Figures 29a-e. Figures 29a-e depict top views of successive branching cross-sections for a single support tree that uses X oniy and Y only branches and results in a total of four support branches from a single support trunk. Figure 29a depicts a single support structure that will be branched into a plurality of structures. This single support structure may be called the "trunk" of the support tree or structure. As will be made clear below, for ease of data manipulation, the trunk can be considered to consist of four separate but identical components which maintain their separate identity, but may be Booleaned together to yield the scanning pattern for any given layer. Of course in practice a real region to be supported might require a plurality of these trunk elements appropπately spaced from each other. Figure 29b depicts a first branching in the X direction. As with the other Figures to follow, the hatched solid regions, as depicted, represent the deposition regions for the instant cross-section whereas the regιon(s) depicted with dashed lines represent the immediately proceeding branch. This way of depicting the deposition regions is done to make the registration between branches clear. This first branching may occur after one or more trunk layers are formed. As with other branches to be descnbed herein after in association with this figure and other figures to follow, branching may extend the dispensed material out from supported regions by a fraction of a pixel, a full pixel, or multiple pixels depending on the drawing order used, the pixel width as compared to the drop width, the number of identical layers to be formed above the present layer (which can compensate for imperfections in the present layer), ability of the mateπal to be partially unsupported, and the like. As with some of the other branches, to be discussed herein after, this branching can be looked at as a two way branch (i.e. one way in the positive X direction and the other way in the negative X-direction) or as a one-way branch of two or more initially overlapped components. As will be seen from the descπption to follow, this first branch may be considered a one-way branching of four initial components wherein two components follow each branching direction. The actual deposition of material from these four components may be based on a Boolean union of the components so that multiple depositions over overlapping regions is avoided. Figure 29c depicts the next branching of the tree wherein this branching may initially occur one or more layers after the branching depicted in Figure 29b. This branching of object components occurs in the same directions as seen in Figure 29b. Figure 29d depicts two branchings in the Y-direction of each of the two branches depicted in Figure 29c. In concept, this may again be considered a single branching in the Y direction of separate components. The branching depicted in Figure 29d is the first branch which begins the separation process of all four components. Figure 29e depicts a final branch for this example embodiment wherein an additional Y- direction branch of each component is made. These final branches can be used to support an object surface as appropπate. If an object surface is located several layers above these final branches, the structures (e.g. columns) of Figure 29e can be extended until the object surface is encountered. If the object surface is not at the same level for all four branches the individual columns or portions of columns can be extended as necessary This extension of support height is similar to other preferred column support embodiments discussed herein and can include the use of bπdge layers and the like Of course if different configurations (e g shapes, positions, and the like) of the four column branched support is desired, modifications (e.g modifications to branching order, branching directions, extension amounts, number of layers between branches, and the like) to the depicted embodiment can be made and will be apparent to one of skill in the art in view of the teachings herein The support trunk depicted in Figure 29a may initially be formed on a previous object cross-section or initial substrate Alternatively, the trunk may begin on top of another support structure such as that depicted in Figure 28a Furthermore, if multiple trees are to be used, branching of the trees may or may not begin on the same layer and may or may not result in each branch being formed after the same number of layers Selection of where to beginning branching and when to make successive branches thereafter, may be based on the geometry of the object to be formed It may be desirable to have the final branching pattem achieved, for a particular tree, several layers before first encounteπng a surface to be supported (e g down-facing object surface) The branching routines performed in association with the example embodiment illustrated in Figures 29a-29e may outlined in the following table Component #1 Component #2 Component #3 Component #4 Build without branching for a desired number of layers (Figure 29a) Branch in the +X Branch in the +X Branch in the -X Branch in the -X direction by desired direction by desired direction by desired direction by desired amount A amount A amount A amount A (Figure 29b) (Figure 29b) (Figure 29b) (Figure 29b) Build without branching for a desired number of layers Branch in the +X Branch in the +X Branch in the -X Branch in the -X direction by desired direction by desired direction by desired direction by desired amount A amount A amount A amount A (Figure 29c) (Figure 29c) (Figure 29c) (Figure 29c) Build without branching for a desired number of layers Branch in the +Y Branch in the -Y Branch in the +Y Branch in the -Y direction by a desired direction by a desired direction by a desired direction by a desired amount A amount A amount A amount A (Figure 29d) (Figure 29d) (Figure 29d) (Figure 29d) Build without branching for a desired number of layers Branch in the +Y Branch in the -Y Branch in the +Y Branch in the -Y direction by a desired direction by a desired direction by a desired direction by a desired amount A amount A amount A amount A Build without branching until a new support style is implemented or until a surface of the object is encountered As desired, the vaπous parameters outlined in the above table can be modified For example the Branching amounts where taken as an amount "A" As appropriate, this amount can vary with different branching levels or it can even vary for different components during the same branching level Figure 30a-30m depict an analogous branching support embodiment to that of Figures 29a- 29e with the exception that the single trunk depicted in Figure 30a will give rise to 16 branches as indicated in Figure 30m For ease of understanding and possibly implementation, the trunk shown in Figure 30a can be considered as consisting of 16 individual but identical components Again, the offset is performed along only one of either the X or the Y directions during a given branching operation for a given component All the considerations noted above in describing Figures 29a-e can be applied to the example embodiment depicted in these Figures as well as the example embodiments to follow Figures 31a-c depicts an additional example embodiment wherein a single trunk, as depicted in Figure 31a, is branched into four elements, as depicted in Figure 31c This embodiment differs from that in Figures 29a - 29c in that branching occurs simultaneously in both the X and Y directions As illustrated, the extent of branching is the same in both the X and Y directions but this extent of branching could be varied between these directions Figure 32a -32d continues the embodiment depicted in Figures 31a- 31c to yield 16 separate branched supports These Figures further illustrate the Structure depicted in Figure 28b wherein two layers for each branch are depicted. In other preferred embodiments other branching pattems are possible For example, instead of yielding rectangular arrays of branched supports from individual trunks, as depicted in the above descnbed examples, hexagonal arrays, tπaπgular arrays, semi-circular arrays, or the like may be formed If the achieved pattems do not fit nicely together, it may be desirable to use a mixture of pattems which are alternated in an appropriate fashion to give a good fitting or meshing of the final support structures such that a down-facing surface can be adeαuately supported Other preferred embodiments may use multiple trunks for supporting single groups of branching supports It is anticipated that these branching support embodiments might yield better down-facing surface than achieved with some of the other preferred embodiments since it is believed that the final support structures that contact the object will be more uniformly spaced As noted above, the branched support embodiments described herein might be a part of a larger support structure or hybrid support structure. Other modifications to the above embodiments will be apparent to one of skill in the art after studying the teachings herein If the geometry and direction sensitive interlacing techniques described above are used it may be possible to build smaller diameter and/or more closely spaced structures to provide a better working surface while still providing reasonable vertical accumulation rates In the preferred embodiment, deposited drop diameter is approximately the same as the preferred pixel diameter (about 2.9 - 34 mils). In general, however, the pixel separation between supports (e.g. separation between support columns) is less critical than the separation relative to the falling drop diameter (e.g. 2 mils) and impacted (or deposited) drop diameter Preferably the horizontal spacing between supports (e g support columns) is less than 6 drop diameters on the layer immediately preceding the layer containing the down-facing surface to be supported More preferably, the spacing is less than 3 falling drop diameters, and most preferably, the spacing is less than 1 to 2 falling drop diameters It has been found useful to include periodic bπdging elements between the support columns to limit their ability to shift from their desired XY positions as they grow in height Typically the smaller the diameter of the support columns the more often bridging elements or layers are needed These bndging elements may extend one or more layers in height In the preferred embodiment, it has been found that a single layer (1-2 mils) of bridging elements is not completely effective and that more than five layers (5-10 mils) makes the overall support structure too rigid Thus when using the preferred 3 by 3 pixel supports, bridging layers are preferrably between 2 layers (2-4 mils) and 5 layers (5-10 mils) in height and most preferrably 3 layers (3-6 mils) in height Furthermore, it has been found that the bndging layers are preferrably repeated every 75 mils to 2 inches more preferably every 100 to 300 mils, and most preferrably every 100 to 200 mils For use with other materials, building parameters, or building conditions formation and analysis of test parts can be used to determine the effective bridge thickness and separations thicknesses When bridging layers are periodically used they may bind all support columns together or they may bind only a portion of them together wherein the other columns were bound on a previous use of bπdging or will be bound on a subsequent use of bridging In other words, the bridging elements may form a solid plane of deposited material or alternatively they may form only a partially solid plane (e g a checkboard pattern) which connects some of the columns together The support columns may or may not be shifted from their previous XY positions when they are restarted after formation of bndging layers Another preferred support structure which emphasizes easy removal and good down-facing surface finish over speed of object production is known as a checkboard support The cross-sectional configuration of this support structure is depicted in Figure 14 Along each raster line, deposition occurs using every other pixel (300 pixels/inch) and in adjacent raster lines the deposition pixels are shifted along the line by one pixel width One preferred version of this support does not use ID oveφπnting, but can use DD overprinting or multiple exposures to increase deposition per layer Without DD overprinting or multiple exposures, the layer thickness when using this type of support in the preferred embodiment is limited to under 0 4 to 0.5 mils, instead of the approximately 1 3 mils obtainable with some preferred embodiments described previously Instead of using DD oveφπntiπg or multiple exposures with these supports it is possible to not use the preferred ID overprinting of the object, and simply deposit material in thinner layers (e g 0 3 to 0 5 mils per layer) Overpπnting of the object does not need to be utilized as the extra mateπal would simply need to be removed during the planaπzation ste. Since raster scanning is used and since the speed of forming a layer is the same with or without oveφπnting, build styles according to these techniques are approximately 3 to 4 times slower than equivalent build styles where four times oveφnnt g are used Though there is a significant increase in build time the improvement in surface finish may warrant its use under certain circumstances When building checkerboard supports, regular use of bridging layers is preferred (e g every 30 to 100 mils of z-height) to ensure column integrity The bridging layers should compπse a sufficient number of layers to ensure their effectiveness (e g about the same thickness of the bndging layers discussed above) A drop-on/drop-off checkerboard pattern (in terms of drop width) is where the solidified elements are 1 drop wide (deposition width), and spacing between the center points of successive elements is greater than 1 drop width but less than 2 drop widths Line supports (in terms of drop width) comprise line elements which are approximately one impacted droplet diameter in width where the spacing between elements tangential to the orientation of the lines is less than 1 drop width (i e overlapping), while the spacing between elements perpendicular to the line orientation is greater than 1 drop width Preferably, the spacing between elements perpendicular to the line orientation is also less than 2 drop widths N-by-N column supports (in terms of pixels; are N-on, preferably one or two-off in the main scan direction and N-on, and preferably 1-off in the index direction The width of the columns and spacing therebetween can be calculated based on a knowledge of the pixel spacing, the drop diameter and any overprinting used The preferred spacing between deposited matenal in adjacent columns is under one to two droplet diameters Another possible support style involves the use of solid or periodically broken lines which are preferably less than 3 pixels wide (less than 10 mils) and more preferably 1 to 2 pixels or less in width (less than 3 3 to 6 6 mils) and are separated by 1 to 2 pixels or less of uπdeposited material (less than 3 3 to 66 mils) These supports may run along the the mam scanning directions, secondary scanning directions, or other directions Another type of support is curved line supports which follow the boundary of an object Alternatively, the support pattern can differ at different areas of the cross- section It can also be displaced from the boundary of the object by N pixels (or drop widths) in the scan direction, or M pixels (or drop widths) in the index direction Some other alternatives involve building supports from a different material than used to form the surface or boundary regions of the object Other alternatives might use a different support mateπal only on one or more of the layers adjacent to the object Hybrid Supports: Further types of support structures useful for Selective Deposition Modeling are Hybπd Supports In its simplest sense, a hybird support is a support structure that includes at least two different types of support structures Preferably, the structures used in a hybnd support vary depending on the height of the support and more particularly, the structure at any given point may depend on the distance from that point to an up-facing and/or down-facing surface of the object In otherwords, the support structures are tailored to the most appropriate structure based on the distance to the object In an exemplary embodiment, the support pattem is changed when the point is located a predetermined number of layers (e g , 4 to 9) below a down-facing surface In another, the drop density per unit area or drop density ratio (defined as the drops to non-drops per unit area ratio) of the supports is decreased as a down-facing surface is approached In a variant of these embodiments one or more layers of shelving (or intermediate) layers are used when transitioning from higher to lower drop density ratio support structures In still another exemplary embodiment, the drop density ratio is increased as an up-facing surface is left (e g , 4 or more layers away from an up-facing surface) In an optional variant of this embodiment one or more layers of shelving (intermediate) layers are used when transitioning from lower to higher drop density ratio support structures It is also conceivable that support structures could vary not just based on vertical distance from the object but also based on horizontal distance as well For example, when horizontally bordering the object a different type of support may be useful than when some distance away from the object An example Hybrid support is depicted from the side in Figure 20 As shown the structure extends from surface 23 which may be the building platform or which may be an up-facing surface of the object being built, to support down-facing surface 24 As depicted, the support structure consists of five components (1) thin fiber-like columns 25 which contact surface 23 (if surface 23 is not an up- facing surface of the object this component of the support structure can be eliminated), (2) more massive columns 26 situated on top of the fiber-like columns 25, (3) intermediate layers 27 (i e a final bπdging layer), (4) thin, fiber-like columns 28 situated on top of the intermediate layers and which directly contact down-facing surface 24 and (5) bridging layers 29 which are used to fuse two or more of the massive columns together and which are distπbuted at various places amongst the columns 26 The thin columns 25 and 28 are both 1 pixel in cross-section (3 3 x 3 3 mils) and form a "checkerboard" pattem as shown in Figure 14a The result is a series of thin, fiber-like columns which are spaced by 1 pixel from adjacent columns, and which easily separate from surfaces 23 and 24 These are equivalent to the checkerboard supports discussed above Based on the one-pixel on one- pixel off deposition pattern of these supports the drop density ratio is approximately 1 If the support does not start on an up-facing surface of the object columns 25 can be skipped Columns 25 and 28 should be between 3 mils and 15 mils in height and preferably about 4-6 mils in height The height should be held to a minimum since it is desired that these supports be used in combination with an object which is being formed with 4 times ID overprinting and since when using a single pass on these support structures without overpπnting they accumulate at a much slower rate than the object On the other hand, it is desired that these supports have some height since the needlelike elements tend to melt down when the down-facing surface of the object is dispensed onto them Columns 26 are 3 x 3 pixels in cross-section (9 9 mils x 9 9 mils), and are spaced 2 pixels from adjacent columns in the scanning direction, and 1 pixel from adjacent columns in the index direction These column supports are equivalent to the most preferred supports discussed above As discussed above, the primary reason for the extra space in the mam scanning direction is the fact that these supports will receive 4 times oveφπnting The cross-sectional pattern formed by these columns is shown in Figure 15 and 18 The result is a series of columns more massive than fiber-like columns 25 and 28 These columns, unlike the others, can be arbitrarily tall The reason is that the larger cross- sections of these columns allow the columns to grow at about the same rate as the part itself (about 1 3 mils/layer) As previously discussed, it is preferred that bridges 29 be used to fuse adjacent ones of columns 26 together peπodically to prevent "wandeπng" of these columns which can occur after building up for some distance The spacing of the bridges is preferably in the previously discussed range The intermediate layers 27 represent an optional final layer of bridging which can function as a transition between columns 26 and columns 28 The reason a transition layer is useful is that the columns 28 are about the same size or smaller than the spaces between the columns 26, with the result that without the transition layers, the columns 28 may fall into these spaces In one preferred approach, intermediate layers as a whole would not be used and instead careful placement of columns 28 on top of columns 26 would occur or only the necessary portions of intermediate layers 27 would be used Preferrably, if used these intermediate layers are of similar thickness to that of the previously discussed bridging layers It should be appreciated that intermediate layers are not needed between columns 25 and columns 26, because the columns 26 are larger in cross-section than the spacing between the columns 25 Accordingly, these larger columns can be built directly on top of the smaller columns without the need of intermediate layers Other hybrid supports are possible that make other combinations with the previously descnbed support elements The hybrid and other support structures may also be used to form intemal portions of objects Additional alternatives exist for building supports For example, it is also possible to build the support from a mateπal which is different from that used to build the part Another possibility is to add a fluid such as water between the interstices of the above described support structures in order to provide additional support and also for aiding heat dissipation In such an approach, it is advantageous to use a fluid that has a greater density than the building material That will give buoyancy to the drops of building material that fall between the interstices of the columns The mateπal should also be chosen such that its surface energy is matched to that of the building mateπal in order to prevent a meniscus forming between the fluid and the columns An example of such a mateπal is a surfactant Another possibility is to shoot air jets upward between the interstices of the columns In this approach a heat dissipation effect and buoyancy are possible Another possibility is to fill the interstices of a reduced number of column supports (e g columns placed 0 1 to 1 inch or more apart) with particles Moreover, such particles could be formed from the building material by allowing or causing the droplets to solidify before they reach the working surface (such as by increasing the distance between the dispensing head and the working surface), or by coating the droplets before they land with a mateπal that sublimes, i e , goes directly from a solid to a gas Supports preferably space the object from 50 to 300 mils from the surface of the building platform Alternatively, the object may be directly built on the platform In this alternative, the platform may be covered with a flexible sheet material that will allow easy separation of the object from the πgid platform and then from the sheet material An electric knife may be used to separate the supports from the platform in which case it is preferred that object be placed 150 to 300 mils above the platform surface A thin comb-like device with long teeth has been found effective for removing the supports from the platform In this case the thickness of the device dictates the required spacing between the object and the platform, typically between 50 and 200 mils The supports may be removed from the object by light rubbing, brushing or by use of a small probe device such as a dental tool Another variant involves incorporating the subject embodiments into an integrated system, which includes a capability for automatic part removal, and a cooling station Other alternatives involve using a low-melting, point metal as a building material, a material filler, or different materials on different raster lines or drop locations Further alternatives involve using larger drops for support building than for part building Another alternative involves the use of powdered supports which may be formed by allowing or causing the droplets to solidify before they reach the working surface as described above Other embodiments might build up objects based on different main-scanning direction orientations (e g Y or Z), other secondary scanning direction orientations (e g X or Z) and other stacking orientations (e g X or Y) Other embodiments might use other absolute movement schemes to achieved the desired relative movements between the object and print head For example in some embodiments absolute movement of the print head might occur in all three directions, while in other embodiments absolute object movement might occur in all three-directions In still other embodiments, non-Cartesian movement of the print head or object might be used and jetting directions may vary from layer to layer or portion of layer to portion of layer Though some embodiments have been described under headings inserted in the application these embodiments should not be considered as pertaining only to the topic indicated by the header Furthermore though headers were used to enhance the readability of this specification, all disclosure relevant to the particular topic recited by the header should not be considered as falling within those single sections All embodiments disclosed herein are useful separately or in combination with other embodiments disclosed herein While embodiments and applications of this invention have been shown and descnbed, it will be apparent to those skilled in the art that many more modifications are possible without departing from the inventive concepts herein The invention therefore, is not to be restricted except in the spirit of the appended claims Appendix A Tables l-lll Detailing preferred mateπals for use in some preferred embodiments Table I describes the component formulations COMPONENT A paraffin wax. melting point 60 °C B pure monomer hydrocaΦon resm comprising a copolymer of α-methylstyrene, vinyl toluene viscosity 1000 cps @130°C, softening point 78-95°C C medium hard microcrystalline ester wax, viscosity 16 cps @ 100"C; melting point 78-86°C, penetration 1 7 mm @ 25°C D hard microcrystalline wax, viscosity 16 cps @ 100°C, melting point 93°C, penetration 0 55mm @ 25°C E soft microcrystalline wax, viscosity 13 cps @ 100°C, melting point 69 °C, penetration 2.9mm @ 25°C F ethylene-vinyl acetate copolymer viscosity 575 cps @ 140°C, melting point 92°C G tetrakisfmethylene (3,5-dι-tert-butyl-4-hydroxyhydrocιnnamate)] methane, an antioxidant; melting point 110-125°C H dioctylphthalate (DOP), a plasticizer 1 dioctylterephthalate (DOTP), a plasticizer J 2,6-dι-tert-butyl-4-methylphenal, an antioxidant K dnsononyl phthalate (DINP), a plasticizer L dye or coloπng compound M methacrylate terminated polystyrene N free radical photoinitiator 0 polyethylene oxide, average molecular weight 2000 P epoxy novolac oligomer Q catomc photoinitiator R multifunctional acrylate Appendix A Tables l-lll Detailing preferred materials for use in some preferred embodiments Table ll-A and ll-B present formulations of preferred materials for use in some preferred embodiments by component designation TABLE ll-A • Weight % COMPONENT Ex. 1 Ex. 2 Ex. 3 Ex.4 Ex. 5 Ex. 6 A 21 21 44 25 20 20 B 49 49 25 20 25 50 C 12 12 12 - - 12 D 5 5 6 10 5 5 E 5 5 6 20 20 5 F 4 4 2.3 20 25 4 G 2 2 - - - - H - 2 - - - 1 - - 2.3 2 2.5 2 J - - 2.3 3 2.5 2 K 2 - - - - - PROPERTIES Viscosity @ 130 cps /°C 22 22 7 20 28 24 °C Hardness shore D 28 26 Impact Energy kJ/m 2 1 4 2.2 Appendix A: Tables l-lll Detailing preferred materials for use in some preferred embodiments TABLE ll-B COMPONENT Ex. 7 Ex. 8 Ex. 9 A 21 M 51 C 12 D 5 E 5 F 5 N 1 1 0 77 77 P 22 Q 1 R 22 Appendix A Tables l-lll Detailing preferred materials for use in some preferred embodiments TASLE I COMPONENT NAME SUPPLIER A Paraffin Allied Signal Inc., Momstown, NJ B "PICOTEX LC" Hercules, Inc., Wilmington, DE C "X-22" Hoechst Celanese Corp., Somerville, NJ D "C-700" Petrolite Corp., Tulsa. OK E "ULTRAFLEX" Petrolite Corp , Tulsa. OK F "AC-400" Allied Signal Inc., Momstown, NJ G "IRGANOX 1010" Ciba-Geigy Corp., Hawthorne, NY H DOTP Aldrich Chemical Co., Inc., Milwaukee, Wl 1 DOP Aldrich Chemical Co., Inc., Milwaukee, Wl J "BHT" Aldπch Chemical Co., Inc., Milwaukee, Wl K DINP Aπstech Chemical Corp., Cleveland, OH M "13K-RC" Sartomer Co., West Chester, PA N "IRGACURE 184" Ciba-Geigy Corp., Hawthorne. NY 0 Polyethelene Oxide Dow Chemicals. Midland, Ml P "DEN 438" Dow Chemicals. Midland, Ml Q "UVI 6974" Union Carbide Chemicals, Danbury, CT R "SR 399" Sartomer Co.. West Chester. PA S P E C i E i C A I I O N METHOD AND APPARATUS FOR DATA MANIPULATION AND SYSTEM CONTROL IN A SELECTIVE DEPOSITION MODELING SYSTEM This application is a continuation-in-part of U S patent application 08/534 447 filed September 27, 1996, now pending 1 Field of the Invention This invention relates to techniques for data manipulation and building control for forming three-dimensional (3D) objects more particularly it relates to techniques for use in Rapid Prototyping and Manufacturing (RP&M) Systems and most particularly to data manipulation and building control methods and apparatus for use in a Thermal Stereolithography (TSL) system Fused Deposition Modeling (FDM) system or other Selective Deposition Modeling (SDM) system 2 Background Infomiation Vaπous approaches to automated or semi-automated three-dimensional object production or Rapid Prototyping & Manufacturing have become available in recent years, characterized in that each proceeds by building up 3D objects from 3D computer data descriptive of the objects in an additive manner from a plurality of formed and adhered laminae These laminae are sometimes called object cross-sections, layers of structure object layers, layers of the object or simply layers (if the context makes it clear that solidified structure of appropπate shape is being referred to) Each lamina represents a cross-section of the three-dimensional object Typically lamina are formed and adhered to a stack of previously formed and adhered laminae In some RP&M technologies, techniques have been proposed which deviate from a stπct layer-by-layer build up process wherein only a portion of an initial lamina is formed and pπor to the remaining portιon(s) of the initial lamina at least one subsequent lamina is at least partially formed According to one such approach, a three-dimensional object is built up by applying successive layers of unsolidified flowable material to a working surface, and then selectively exposing the layers to synergistic stimulation in desired pattems, causing the layers to selectively harden into object laminae which adhere to previously-formed object laminae In this approach, mateπal is applied to the working surface both to areas which will not become part of an object lamina, and to areas which will become part of an object lamina Typical of this approach is Stereolithography (SL), as descnbed in U S Patent No 4,575,330, to Hull According to one embodiment of Stereolithography, the synergistic stimulation is radiation from a UV laser, and the material is a photopolymer Another example of this approach is Selective Laser Sintering (SLS), as descnbed in U S Patent No 4,863,538, to Deckard, in which the synergistic stimulation is IR radiation from a C0 2 laser and the material is a sinterable powder A third example is Three-dimensional Pπnting (3DP) and Direct Shell Production Casting (DSPC), as descnbed in U S Patent Nos 5,340,656 and 5,204,055 , to Sachs, et al , in which the synergistic stimulation is a chemical binder and the material is a powder consisting of particles which bind together upon selective application of a chemical binder According to a second such approach, an object is formed by successively cutting object cross-sections having desired shapes and sizes out of sheets of material to form object lamina Typically in practice, the sheets of paper are stacked and adhered to previously cut sheets pπor to their being cut but cutting prior to stacking and adhesion is possible Typical of this approach is Laminated Object Manufacturing (LOM), as described in U S Patent No 4,752,352. to Feygin in which the mateπal is paper, and the means for cutting the sheets into the desired shapes and sizes is a C0 2 laser U S Patent 5,015,312 to Kinzie also addresses LOM According to a third such approach object laminae are formed by selectively depositing an unsolidified flowable material onto a working surface in desired patterns in areas which will become part of an object laminae After or during selective deposition, the selectively deposited mateπal is solidified to form a subsequent object lamina which is adhered to the previously-formed and stacked object laminae These steps are then repeated to successively build up the object lamma-by-lamina This object formation technique may be generically called Selective Deposition Modeling (SDM) The main difference between this approach and the first approach is that the material is selectively deposited only in those areas which will become part of an object lamina Typical of this approach is Fused Deposition Modeling (FDM), as described in U S Patent Nos 5,121 ,329 and 5.340,433, to Crump, in which the material is dispensed while in a flowable state into an environment which is at a temperature below the flowable temperature of the material, and which then hardens after being allowed to cool A second example is the technology descnbed in U S Patent No 5,260,009 to Penn A third is Ballistic Particle Manufacturing (BPM), as described in U S Patent Nos 4,665,492, 5,134,569, and 5,216,616, to Masters in which particles are directed to specific locations to form object cross-sections A fourth example is Thermal Stereolithography (TSL) as descnbed in U S Patent No 5 141 ,680, to Almquist et al When using SDM (as well as other RP&M building techniαues) the appropπateness of vanous methods and apparatus for production of useful objects depenαs on a number of factors As these factors cannot typically be optimized simultaneously, a selection of an appropriate building technique and associated method and apparatus involve trade offs depending on specific needs and circumstances Some factors to be considered may include 1 ) equipment cost, 2) operation cost, 3) production speed, 4) object accuracy, 5) object surface finish, 6) material properties of formed objects, 7) anticipated use of objects, 8) availability of secondary processes for obtaining different mateπal properties, 9) ease of use and operator constraints, 10) required or desired operation environment, 11) safety, and 12) post processing time and effort In this regard there has been a long existing need to simultaneously optimize as many of these parameters as possible to more effectively build three-dimensional objects As a first example, there has been a need to enhance object production speed and lower set up time and file preparation time when building objects using a Selective Deposition Modeling technique (SDM) as descnbed above (e g Thermal Stereolithography) while simultaneously maintaining or reducing the equipment cost A cntical problem in this regard has been the need for an efficient technique for generating and handling build data. Another cntical problem involves the need for an efficient technique for generatang support data appropπate for supporting an object during formation Additional problems involve the existence of control software which is capable of manipulating the massive amounts of data involved in real time, of compensating for jet misfiπng or malfunctioning, of adjusting data so it is accessible in the order needed, and for efficiently providing geometry sensitive build styles and deposition techniques Appropπate build styles and support structures for use in SDM for which a data generation technique is needed are described in U S Patent Application No 08/534,813 Accordingly, there is a long-felt but unmet need for methods and apparatus to derive data and control an SDM system to overcome the disadvantages of the prior art All patents referred to in this section of the specification are hereby incoφorated by reference as if set forth in full 3 Related Patents Applications The following applications are hereby incorporated herein by reference as if set forth in full herein Filing Date Application Title Status No 9/27/95 08/534,813 Selective Deposition Modeling Pending Method and Apparatus for Forming Three-dimensional Objects and Supports 9/27/95 08/534,447 Method and Apparatus for Data Pending Manipulation and System Control in a Selective Deposition Modeling System 9/27/95 08/535,772 Selective Deposition Modeling Pending Materials and Method 9/27/95 08/534 477 Selective Deposition Modeling Pending Method and System The assignee of the subject application, 3D Systems, I ic , is filing this application concurrently with the following related application, which is incorporated by reference herein as though set forth in full Docket No Filing Date Application Title Status No USA.142 Concurrently Unknown Selective Deposition Modeling Pending herewith Method and Apparatus for Forming Three-dimensional Objects and Supports According to Thermal Stereolithography and some Fused Deposition Modeling techniques, a three- dimensional object is built up layer by layer from a material which is heated until it is flowable, and which is then dispensed with a dispenser The matenal may be dispensed as a semi-continuous flow of mateπal from the dispenser or it may alternatively be dispensed as individual droplets In the case where the mateπal is dispensed as a semi-continuous flow it is conceivable that less stππgent working surface cπteπa may be acceptable An early embodiment of Thermal Stereolithography is descnbed in U S Patent No. 5,141 ,680 which is hereby incorporated by reference. Thermal Stereolithography is particularly suitable for use in an office environment because of its ability to use non-reactive, non¬ toxic mateπals Moreover, the process of forming objects using these mateπals need not involve the use of radiations (e g UV radiation IR radiation, visible light and/or laser radiation), heating mateπals to combustible temperatures (e g burning the material along cross-section boundaπes as in some LOM techniques), reactive chemicals (e g monomers, photopolymers) or toxic chemicals (e g solvents), complicated cutting machinery, and the like, which can be noisy or pose significant πsks if mishandled Instead, object formation is achieved by heating the material to a flowable temperature then selectively dispensing the material and allowing it to cool U S Patent Application No 08/534 813 is directed primarily to Build and Support styles and structures which can be used in a preferred Selective Deposition Modeling (SDM) system based on TSL principles Alternative build and support styles and structures are also described for use in other SDM systems as well as for use in other RP&M systems U S Patent Application No 08/535 772 is directed to the preferred material used by the preferred SDM/TSL system descπoed hereinafter Some alternative materials and methods are also descnbed U S Patent Application No 08/534 447 is a parent application for the instant application and is directed to data transformation techniques for use in converting 3D object data into support and object data for use in a preferred Selective Deposition Modeling (SDM) system based on TSL (thermal stereolithography) pπnciples This referenced application is also directed to vaπous data handling, data control, and system control techniques for controlling the preferred SDM/TSL system descnbed hereinafter Alternative data manipulation techniques and control techniques are also descnbed for use in SDM systems as well as for use in other RP&M systems The assignee of the instant application, 3D Systems, Inc is also the owner of a number of other U S Patent Applications and U S Patents in RP&M field and particularly in the Stereolithography portion of that field The following commonly owned U S Patent Applications and U S Patents are hereby incoφorated by reference as if set forth in full herein App No Topic Status and/or Patent No 08/148 544 Fundamental elements of Thermal Stereolithography are 5,501 ,824 described 08/484,582 Fundamental elements of Stereolithography are taught Pending 08/475,715 Vaπous recoating techniques for use in SL are descnbed Pending including a mateπal dispenser that allows for selective deposition from a plurality of orifices 08/479,875 Vanous LOM type building techniques are described Pending /486.098 A description of curl distortion is provide along with various Pending techniques for reducing this distortion /475,730 A descπption of a 3D data slicing technique for obtaining Pending cross-sectional data is described which utilizes boolean layer comparisons to define down-facing, up-facing and continuing regions Techniques for performing cure-width compensation and for producing various object configurations relative to an initial CAD design are also descnbed /480,670 A descπption of an early SL Slicing technique is described Pending including vector generation and cure width compensation /428,950 Various building techniques for use in SL are described Pending including various build styles involving altemate sequencing, vector interlacing and vector offsetting for forming semi-solid and solid objects /428 951 Simultaneously multiple layer cuπng techniques for SL are Pending taught including techniques for vertically comparing regions and correcting errors due to over curing in the z-direction In additional horizontal comparison techniques are discussed for defining horizontally distinct regions including the use of erosion routines /405,812 SL recoating techniques using vibrational energy are Pending described /402,553 SL recoating techniques using a doctor blade and liquid Pending level control techniques are described /382.268 Several SL recoating techniques are descnbed including Pending techniques involving the use of ink jets to selectively dispense mateπal for forming a next layer of unsolidified mateπal /182,801 Support structures for SL are descnbed 4,999,143 /183,015 Placement of holes in objects for reducing stress in SL 5,015,424 objects are described /365,444 Integrated SL building, cleaning and post curing techniques 5,143,663 are descnbed /824,819 Vaπous aspects of a large SL apparatus are descnbed 5,182,715 /605,979 Techniques for enhancing surface finish of SL objects are 5,209,878 descnbed including the use of thin fill layers in combination with thicker structural layers and meniscus smoothing /929,463 Powder coating techniques are descnbed for enhancing 5,234,636 surface finish /939,549 Building techniques for reducing curl distortion in SL by 5,238,639 balancing regions of stress and shnnkage SUMMARY OF THE INVENTION The instant invention embodies a number of techniques (methods and apparatus) that can be used alone or in combination to address a number of problems associated with data generation data handling and system control for use in forming 3D objects by Selective Deposition Modeling Though pnmaπly directed to SDM techniques the techniques described hereinafter can be applied in a vaπety of ways to the other RP&M technologies as described above to enhance system throughput by providing enhanced data manipulation and generation techniques Furthermore, the techniques described herein can be applied to SDM systems that use one or more building and/or support materials wherein one or more of the materials are selectively dispensed, wherein others may be dispensed non-selectively, and wherein elevated temperatures may or may not be used for all or part of the materials to aid in their selectively deposition The techniques can be applied to SDM systems wherein the building mateπal (e g paint or ink) is made flowable for dispensing purposes by adding to it a solvent (e g water, alcohol acetone, paint thinner or other solvents appropriate for specific building materials), which material can be made to solidify after dispensing by causing the removal of the solvent (e g by heating the dispensed material, by dispensing the matenal into a partially evacuated (i e vacuum) building chamber, or by simply allowing sufficient time for the solvent to evaporate) Alternatively and/or additionally, the building material (e g paint) may be thixotropic in nature wherein an increase in shear force on the material could be used to aid its dispensing or the thixotropic property may simply be used to aid the mateπal in holding its shape after being dispensed Alternatively and/or additionally, the material may be reactive in nature (e g a photopolymer, a thermal polymer, a one or two-part epoxy material, a combination mateπal such as one of the previously mentioned matenals in a combination with a wax or thermal plastic mateπal) or at least solidifiable when combined with another material (e g plaster of pans & water) wherein after dispensing the material is reacted by appropriate application of prescπbed stimulation (e g heat, EM radiation ( visible IR, UV x-rays, etc ) a reactive chemical, the second part of a two part epoxy the second or multiple part of a combination) wherein the building material and/or combination of materials become solidified Of course. Thermal Stereolithographic materials and dispensing techniques may be used alone or in combination with the above alternatives Furthermore, vaπous dispensing techniques may be used such as dispensing by single or multiple ink jet devices including hot melt ink jets, bubble jets, etc and continuous or semi-continuous flow, single or multiple oπfice extrusion nozzles or heads A first object of the invention to provide a method and apparatus for converting three- dimensional object data into cross-sectional data A second object of the invention is to provide a method and apparatus for production of objects including a method and apparatus for converting three-dimensional object data into cross- sectional data A third object of the invention to provide a method and apparatus for obtaining support data from three-dimensional object data A fourth object of the invention is to provide a method and apparatus for production of objects including a method and apparatus for obtaining support data and using the support data duπng object formation. It is intended that the above objects can be achieved separately by different aspects of the invention and that additional objects of the invention will involve various combinations of the above independent objects such that synergistic benefits may be obtained from combined techniques. Other objects of the invention will be apparent from the descπption herein. BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 is a diagram of a preferred Thermal Stereolithography system; Figure 2a and 2b illustrate the orifice plate of the pnnt head of Figure 1 at two different angles. Figure 3 is a more detailed drawing of the plananzer of Figure 1 , Figure 4 illustrates the relative spacing between adjacent nozzles on the orifice plate and adjacent raster lines, Figure 5 illustrates the grid of pixels which defines the data resolution of the system: Figures 6a - 6d illustrates several overpπnting schemes, Figure 7 illustrates a first embodiment of the subject invention; Figure 8a-8b illustrates intersecting an .STL file with slicing planes; Figure 9 illustrates the effect of the boolean extraction operation; Figure 10 illustrates a second embodiment of the subject invention; Figure 11a illustrates the ordering of triangles in the z-direction; Figures 11b-11c illustrate the selection of active tnangles; Figures 12a, 12b, and 12c illustrate altemate forms of representing cross-sectional data; Figures 13a, 13b, and 13c illustrate the placement of transitional data into lists associated with different scan lines; Figures 14-15 illustrate in more detail the boolean extraction operation: Figures 16-17 illustrate the Boolean addition, subtraction, and intersection operations; Figures 18-21 illustrate a two-stage process for generating supports using intermediate layers; Figure 22 illustrates a three-stage process for generating supports; Figures 23-26 illustrate a method of stoπng start/stop data into contiguous words; Figures 27a-27b, 28a-28b. 29a-29b, and 30a-30b illustrate a method of allocating memory to list data representing start/stop transitions; Figure 31 illustrates the property in which successive scan lines are represented by similar numbers of transitions; Figure 32 illustrates the quantization error introduced by the pixeiization of start stop data; Figure 33 illustrates the conversion of RLE data to conversion data; Figure 34a-34c illustrates data for use in building supports; Figure 35a illustrates an assumption about part slope made in the preferred embodiment of the subject invention; Figures 36a-36c illustrate a ring buffer used in the present implementation of the subject invention Figure 37 illustrates a hybrid support structure, Figures 38a-38b illustrate two sample Style types Figures 39a, 39b, and 39c illustrate part support encounters that sometimes occur, Figures 40a-40c represent an example of how style files are used Figures 41a-41f represent additional style types Figures 42a-42e illustrate skewing, Figure 43 illustrates the prescribed sequence of encoder lines, Figure 44 illustrates a resolution problem which can occur through only one firing counter, and Figures 45a-45b illustrate an algorithm for increasing resolution in the scan direction through use of two counters Figure 46a depicts a side view of an object containing a gap along with hypothetical levels and regions upon which formation of different support structures can be based Figure 46b depicts a side view of the object of Figure 46a wherein the gap is filled with various types of support structures Figure 47 depicts the conceptual format of an RLE file DESCRIPTION OF THE PREFERRED EMBODIMENTS As previously discussed, the subject application is directed to data manipulation techniques and system control techniques for implementing support techniques and building techniques appropπate for use in a Selective Deposition Modeling (SDM) system In particular the preferred SDM system is a Thermal Stereolithography (TSL) system The detailed descπption of a preferred embodiment of the invention will begin with a descπption of the preferred TSL system wherein embodiment details will be described as appropπate A more detailed description of preferred building and supporting techniques, preferred mateπai formulations and properties, preferred system and various alternatives are described in previously incorporated U S Patent Applications 08/534,447, 08/535,772, and 08/534 477 Further alternative systems are discussed in a number of the previously incorporated applications and patents especially those referenced as being directly related to or applicable to Selective Deposition Modeling, Thermal Stereolithography or Fused Deposition Modeling As such, the data manipulation techniques and system control techniques hereinafter should be construed as applicable to a variety of SDM, TSL and FDM systems and not inappropriately limited by the examples descnbed herein A preferred apparatus for performing SDM/TSL is illustrated in Figure 1 The apparatus comprises a dispensing platform 18 on which is situated dispensing head 9 (e g multi oπfice inkjet head) and planaπzer 11 The dispensing platform is slidably coupled to X-stage 12 through member 13 The X-stage 12 controUably moves the dispensing platform 18 back and forth in the X-direction, also known as the main scanning direction The motion of the X-stage is under the control of a control computer or microprocessor (not shown) Furthermore, at either side of the platform 18 and/or between the planaπzer 11 and dispensing head 9, fans (not shown) for blowing air vertically down are mounted to help cool the dispensed mateπal and substrate such that the desired building temperature is maintained Of course other mounting schemes for the fans and/or other cooling systems are possible including the use of misting devices for directing vapoπzable liquids (e g water, alcohol, or solvents) onto the surface of the object Cooling systems might involve active or passive techniques for removing heat and may be computer controlled in combination with temperature sensing devices to maintain the dispensed material within the desired building temperature range The dispensing head (or print head) 9 is a commercial pπnt head configured for jetting hot melt inks (e g thermal plastics or wax-like materials), and modified for use in a three-dimensional modeling system wherein the printhead undergoes back and forth movements and accelerations The pπnthead modifications involve configuring any on board reservoir such that the accelerations result in minimal misplacement of material in the reservoir In one preferred embodiment, the head is a 96 jet commercial print head, Model No HDS 96ι, sold by Spectra Corporation, of Nashua, Hew Hampshire including reservoir modifications The print head 9 is supplied material in a flowable state from a Material Packaging & Handling Subsystem (not shown) The material packaging and handling subsystem is described in previously referenced U S Patent Application No 08/534 477 In a preferred implementation all 96 jets on the head are computer controlled to selectively fire droplets through orifice plate 10 when each orifice (i e jet) is appropriately located to dispense droplets onto desired locations In practice, preferrably approximately 12,000 to 16,000 commands per second are sent to each jet selectively commanding each one to fire (dispense a droplet) or not to fire (not to dispense a droplet) depending on jet position and desired locations for mateπal deposition Also, in practice, firing commands are preferrably sent simultaneously to all jets Thus, in a preferred embodiment, the head is computer controlled so as selectively fire the jets so as to simultaneously emit droplets of the molten mateπal through one or more oπfices in onfice plate 10 Of course it will be appreciated that in alternative embodiments, heads with a different numbers of jets can be used, different finng frequencies are possible and, in appropriate circumstances, non-simultaneous finng of the jets is possible The orifice plate 10 is mounted on the dispensing platform 18 such that droplets of mateπal are allowed to emit from the underside of the dispensing platform The oπfice plate 10 is illustrated in Figures 2a and 2b In a preferred embodiment, and as depicted in Figure 2a, the orifice plate (i e. the line of oπfices) is mounted approximately perpendicular to the main scanning direction (e.g X- directioπ) and is configured with N = 96 individually controllable oπfices (labeled 10(1), 10(2), 10(3) . 10(96)) Each dispenser (e g jet) is equipped with a piezoelectπc element which causes a pressure wave to propagate through the mateπal when an electπc firing pulse is applied to the element The pressure wave causes a drop of matenal to be emitted from oπfice The 96 dispensers are controlled by the control computer which controls the rate and timing of the finng pulses applied to the individual dispenser and therefore the rate and timing of droplets being emitted from the oπfices With reference to Figure 2a, the distance "d" between the oπfices in a preferred embodiment is about 8/300 inches (about 26.67 mils or 0677 mm) Thus, with 96 oπfices, the effective length "D" of the oπfice plate is about (N x 8/300 inch) = (96 x 8/300 inches) = 2 56 inches (65 02 mm) One preferred embodiment uses raster scanning to position the print head and orifices to dispense material at desired drop locations. The printing process for each layer is accomplished by a seπes of relative movements between the head and the desired drop locations. Printing typically occurs as the head relatively moves in a main scanning direction. This is followed by a typically smaller increment of movement in a secondary scanning direction while no dispensing occurs, which in turn is followed by a reverse scan in the main scanning direction in which dispensing again occurs. The process of alternating main scans and secondary scans occurs repeatedly until the lamina is completely deposited. Alternative embodiments may perform small secondary scanning movements while main scanning occurs. Because of the typically large difference in net scanning speed along the main and secondary directions such alternatives still result in deposition along nearly perpendicular mam scanning lines (i.e the mam scanning and secondary scanning directions remain substantially perpendicular. Other alternative embodiments may utilize vector scanning techniques or a combination of vector scanning and raster scanning. Other alternative embodiments may use substantially non-perpendicular mam and secondary scanning directions along with algorithms that result in proper placement of droplets. In alternative embodiments the print head may be mounted at a non-perpendicular angle to the main scanning direction. This situation is depicted in Figure 4b wherein the print head is mounted at an angle "α" to the main scanning direction. In this alternative situation the separation between the oπfices is reduced from d to d' = (d x sin α) and the effective length of the print head is reduced to D' = (D x sin α). When the spacing d' is equal to the desired print resolution in the secondary scanning direction (direction approximately perpendicular to the mam scanning direction), the angle a is considered to be the "saber angle". If the spacing d (as when using a preferred embodiment) or d' (as when using some preferred alternative embodiments) is not at the desired secondary pnnt resolution (i.e. the print head is not at the saber angle) then for optimal efficiency in pπnting a layer, the desired resolution must be selected such as to make d or d' an integer multiple of the desired resolution. Similarly, when printing with σ ≠90°, a spacing exists between adjacent jets in the mam scanning direction as well as the secondary scanning direction. This spacing is defined by d" = d x cos α. This spacing in the main scanning direction d", in turn, dictates that optimization of printing efficiency will occur when the desired main pπnt resolution is selected to be an integer multiple of d" ( assuming that firing locations are located in a rectangular grid). This may be alternatively worded by saying that the angle α is selected such that d' and/or d" (preferrably both) when divided by appropriate integers M and P yield the desired main and secondary scanning resolutions. An advantage to using the preferred pπnt head orientation (α = 90°) is that it allows any desired printing resolution in the mam scanning direction while still maintaining optimal efficiency. In other altemative embodiments multiple heads may be used which lay end to end (extend in the secondary scanning direction) and/or which are stacked back to back (stacked in the main scanning direction). When stacked back to back the pnnt heads may have orifices aligned in the main scanning direction so that they print over the same lines or altematively they may be offset from one another so as dispense material along different main scanning Iines. In particular it may be desirable to have the back to back printheads offset from each other in the secondary scanning direction by the desired raster line spacing to minimize the number of main scanning passes that must occur In other alternative embodiments the data defining deposition locations may not be located by pixels defining a rectangular gπd but instead may be located by pixels laid out in some other pattem (e g offset or staggered pattem) More particularly, the deposition locations may be fully or partially varied from layer to layer in order to perform partial pixel drop location offsetting for an entire layer or for a portion of a layer based on the particulars of a region to be jetted Presently preferred pπnting techniques involve deposition of 300, 600 and 1200 drops per inch in the main scanning direction and 300 drops per inch in the secondary scanning direction With reference to Figures 1 and 3, planaπzer 11 is a heated rotating cylinder 18a with a textured (e g knurled) surface Its function is to melt, transfer and remove portions o the previously dispensed layer of material in order to smooth it out, to set a desired thickness for the last formed layer, and to set the net upper surface of the last formed layer to a desired level (i e the desired working surface or working level for forming a next lamina of the object) Numeral 19 identifies a layer of material which has just been deposited by the print head The rotating cylinder 18a is mounted to the dispensing platform such that it is allowed to project from the underside of the platform by a sufficient amount in the Z-direction such that it contacts material 19 at a desired level below the oπfice plate In a preferred embodiment, this amount is set in the range of 0 5 mm to 1 0 mm The rotation of the cylinder 18a sweeps mateπal from the just-deposited layer, identified in the figure with numeral 21, leaving in its wake smooth surface 20 The material 21 adheres to the knurled surface of the cylinder and is displaced until it contacts wiper 22 As shown, wiper 22 is disposed to effectively "scrape" the mateπal 21 from the surface of the cylinder This mateπal, because it is still flowable, is then taken up by the Mateπal Packaging & Handling Subsystem, described in U S Patent Application No 08/534,477, (not shown), whence it is either disposed of or recycled With reference to Figure 1, part-building platform 15 is also provided On this platform 15 is built the three-dimensional object or part identified in the figure with reference numeral 14 This platform 15 is slidably coupled to Y-stage 16a and 16b which controUably moves the platform 15 back and forth in the Y-direction (i e index direction or secondary scanning direction) under computer control The platform is also coupled to Z-stage 17 which controUably moves the platform up and down (typically progressively downward duπng the build process) in the Z-direction under computer control To build a cross-section of a part, the Z-stage is directed to move the part-building platform 15 relative to the pπnt head 9 such that the last-built (i e dispensed and possibly planed) cross-section of the part 14 is situated an appropπate amount below the oπfice plate 10 of the pπnt head The pπnt head in combination with the Y-stage is then caused to sweep one or more times over the XY build region (the head sweeps back and forth in the X direction, while the Y-stage translates the partially formed object in the Y-direction) The combination of the last formed layer of the object and any supports associated therewith define the working surface for deposition of the next lamina and any supports associated therewith Duπng translation in the XY directions, the jets of the pπnt head are fired in a registered manner with previously dispensed layers to deposit mateπal in a desired pattem and sequence for the building of the next lamina of the object Dunng the dispensing process, a portion of the dispensed mateπal is removed by the plananzer in the manner discussed above The X, Y and Z movements, dispensing, and planaπzmg are repeated to build up the object from a plurality of selectively dispensed and adhered layers In an alternative embodiment the step of planarization could be performed independently of the dispensing steps In other alternative embodiments the plananzer may not be used on all layers but instead may be used on selected or peπodic layers As noted previously, in a preferred embodiment the print head is directed to trace a raster pattern An example of this is depicted in Figure 4 As shown, the raster pattern consists of a seπes of raster lines, R(1), R(2), R(N), running in the X-direction or main scanning direction and arrayed along the Y-direction (i e index direction or secondary scanning direction) The raster lines are spaced from one another by a distance d r , which, in a preferred embodiment, is 1/300 inches (about 3 3 mils or about 83 8 μm) Since the orifices of the print head are spaced by the distance d which as discussed is about 26 67 mils (0 6774 μm ) and since the desired number of raster lines may extend in the index direction by a distance greater than the length of the orifice plate, about 2 56 inches (65 02 mm), the print head must be swept over the working surface through multiple passes in order to trace out all desired raster lines This is accomplished by following a two-step process In the first step, the pπnt head is passed 8 times over the working surface, with the Y-stage being indexed by the amount d r after each pass in the main scanning direction In the second step, the Y-stage is indexed by a distance equal to the length of the orifice plate (2 5600 inches + d f (0.0267 inches) = 2 5867 inches (65 70 mm) This two-step process is then repeated until all of the desired raster lines have been traced In other words, a preferred two step process involves a first step of alternating main scanning direction passes with secondary scanning direction movements of an amount equal to the desired raster line resolution until all raster lines between initial lines dispensed by two adjacent jets are scanned Thereafter, a second step including a large index direction increment is made This large index direction increment is equal to the spacing between first and last orifices of the printhead plus one raster iine spacing The first and second steps are repeated until the indexing direction increments, and lines scanned, are sufficient to deposit material on all raster lines required to form the object cross-section (including any necessary supports for forming subsequent cross-sections) In a first pass, for example, the print head might be directed to trace raster lines R(1) (via oπfice 10(1) in Figure 4), R(9) (via oπfice 10(2)), R(17) (via oπfice 10(3)), etc The Y-stage would then be directed to move the building platform the distance d r (one raster line) in the index direction On the next pass, the pπnt head might be directed to trace R(2) (via 10(1)), R(10) (via 10(2)), R(17) (via 10(3)), etc. Six more passes would then be performed with the Y-stage being indexed by the distance d, after each pass, until a total of 8 sweeps have been performed At this time, if there are more raster lines to be traced, the Y-stage would be directed to move the building platform by an amount equal to the full length of the orifice plate + d r , 2 5867 inches (65.70 mm) The two-step process described above would then be repeated until all raster lines have been traced out In alternative embodiments, other Y increments could be made including increments involving both negative and positive movements along the Y-axis This might be done in order to scan raster lines that were initially skipped This will be descnbed further in association with a technique called "interlacing" The firing of the ink jet oπfices is controlled by a rectangular bit map maintained in the control computer or other memory device The bit map consists of a gπd of memory cells, in which each memory cell corresponds to a pixel of the working surface, and in which the rows of the gπd extend in the mam scanning direction (X-direction) and the columns of the gπd extend in the secondary scanning direction (Y-direction) The width of (or distance between) the rows (spacing along the Y- direction) may be different from the width (or length of or distance between ) of the columns (spacing along the X-direction) dictating that different data resolutions may exist along the X and Y directions In alternative embodiments, non-uniform pixel size is possible within a layer or between layers wherein one or both of the pixel width or length is varied by pixel position In other alternatives, other pixel alignment patterns are possible For example, pixels on adjacent rows may be offset in the mam scanning direction by a fractional amount of the spacing between pixels in the main scanning direction so that their center points do not align with the center points of the pixels in the neighboπng rows This fractional amount may be Vi so that their center points are aligned with the pixel boundaπes of adjacent rows It may be 1/3 or some other amount such that two or more intermediate rows of pixels are located between rows where pixels are realigned in the main scanning direction In further alternatives, pixel alignment might be dependent on the geometry of the object or support structure being dispensed For example, it might be desirable to shift pixel alignment when forming a portion of a support pattem that is supposed to bπdge a gap between support columns These and other alternative pixel alignment schemes can be implemented by modifying the pixel configuration or alternatively defining a higher resolution pixel arrangement (in X and/or Y) and using pixel finng pattems that do not fire on every pixel location but instead fire on selected spaced pixel locations which may vary according to a desired random, predetermined or object basis pattem We may define the data resolution in the main scanning direction in teπns of Main Direction Pixels (MDPs) MDPs may be described in terms of pixel length or in terms of number of pixels per unit length In a preferred embodiment MDP = 300 pixels per inch (26 67 mils/pixel or 6774 μm/pixel) In other prefered embodiments MDP = 1200 pixels per inch Similarly the data resolution in the secondary scanning direction may be defined in terms of Secondary Direction Pixels (SDPs) and the SDPs may be descnbed in teπns of pixel width or in terms of number of pixels per unit length In a preferred embodiment SDP = MDP = 300 pixels per inch (26.67 mils/pixel or 6774 μm/prxel) The SDP may or may not be equivalent to spacing between raster lines and the MDP may or may not be equivalent to the spacing between successive drop locations along each raster line The spacing between successive raster lines may be defined as Secondary Drop Locations (SDLs), while spacing between successive drop locations along each raster line may be defined as Main Drop Locations (MDLs) Similar to SDPs and MDPs SDLs and MDLs may be defined in terms of drops per unit length or drop spacing If SDP = SDL there is a one to one correspondence between data and drop locations along the secondary scanning direction and the pixel spacing is equal to that of the raster line spacing. If MDP = MDL there is a one to one correspondence between data and drop locations along the mam scanning direction. If SDL and/or MDL is larger than SDP and MDP, respectively, more drops will need to be fired than that for which data exists, thus each pixel will need to be used in causing more than one droplet to be dispensed. The dispensing of these extra droplets can be done in one of two ways either by dispensing the droplets at intermediate points between the centers of successive pixels (i.e. intermediate dropping, "ID") or alternatively directly on top of pixel centers (i.e. direct dropping, "DD"). In either case this technique is called "overprinting" and results in faster build up of mateπal and eases mechanical design constraints involving maximum scan speeds and acceleration rates since the same Z-build up can occur while moving the printhead and/or object more slowly The difference in ID oveφπnting versus non-overpπnting or DD overprinting is depicted in Figures 6a to 6d. Figure 6a depicts a single drop 60 being deposited and an associated solidified region 62 surrounding it when the print head is moving in direction 64 On the other hand, Figure 6b depicts the same region being cured but using the ID overpnnting technique where two drops 60 and 66 are deposited in association with the single data point when the head is moving in direction 64 The deposition zone filled by the two drops is depicted by region 68 Figure 6c shows a similar situation for a four drop ID overprinting scheme wherein the drops are indicated by numerals 60, 70, 66 and 72 and the deposition zone is depicted by 76 and wherein the scanning direction is still depicted by numeral 64 Figure 6d depicts a similar situation for a line of pixels 78, 80, 82, 84, 86 and 88, wherein numeral 90 depicts the length of the deposition zone without overpπnting and the numeral 92 depicts the length of the deposition zone when using a four drop ID overprinting technique. The above can be generalized by saying that ID oveφππting adds from approximately Vi to just under 1 additional pixel length to any region wherein it is used. Of course, the more overprinting drops that are used, the more vertical growth a pixel region will have. If SDL and/or MDL is less than SDP and/or MDP, respectfully, drops will be fired at fewer locations than those for which data exists, at least for a given pass of the print head. This data situation may be used to implement the offset pixel and/or non-uniform sized pixel techniques discussed above. An N row by M column grid is depicted in Figure 5. As shown, the rows in the gπd are labeled as R(1), R(2), . . , R(N), while the columns in the grid are labeled as C(1), C(2), . , C(M). Also shown are the pixels making up the grid. These are labeled as P(1 ,1), P(1 ,2), . . ,P(M,N) . To build a cross-section, the bit map is first loaded with data representative of the desired cross-section (as well as any supports which are desired to be built). Assuming, as with the preferred embodiment, a single build and support material is being used, if it is desired to deposit mateπal at a given pixel location, then the memory cell corresponding to that location is appropπately flagged (e.g. loaded with a binary "1") and if no mateπal is to be deposited an opposite flag is used (e.g. a binary "0"). If multiple mateπals are used, cells corresponding to deposition sites are flagged appropnately to indicate not only drop location sites but also the mateπal type to be deposited For ease of data handling, compressed data defining an object or support region (e g on-off location points aiong each raster line) can be booleaned with a fill pattern descπption to be used for the particular region to deπve a final bit map representation used for finng the dispensing jets The raster lines making up the gπd are then assigned to individual orifices in the manner described earlier Then, a particular oπfice is directed to fire or not over a pixel depending on how the corresponding cell in the bit map is flagged As discussed above the pπnt head is capable of depositing droplets at many different resolutions In the preferred embodiments of the present invention SDP = SDL = 300 pixels and drops per inch However MDP is allowed to take on three values in the preferred embodiment 1) MDL = 300 drops per inch and MDP = 300 pixels per inch, 2) MDL = 600 drops per inch and MDP= 300 pixels per inch or 3) MDL = 1200 drops per inch and MDP = 300 pixels peπnch When the MDL to MDP ratio is greater than one, the extra drops per pixel are made to occur at intermediate locations (ID oveφπnting) between the centers of the pixels With the currently preferred pπnt head and mateπal, the volume per drop is about 100 picoliters which yields drops roughly having a 2 mil (50 8 μm) diameter With the currently preferred print head the maximum frequency of firing is about 20 Khz By way of companson a finng rate of 1200 dpi at 13 ips involves a frequency of 16 Khz, which is within the permissible limit A first preferred embodiment for producing data appropriate for part building in a Selective Deposition Modeling system (e g a Thermal Stereolithography System), including generatang data representative of supports is illustrated in Figure 7 As shown, the method begins by using the Boolean Layer Slice process (represented by module 31) to convert STL file 30 into SLI file 32 The Boolean Layer Slice process, as well as the STL and SLI formats, are descnbed in the above referenced U S Patents and Applications, (e g U S Patent Application No 08/475 730 (hereafter • 730)) The SLI file is then input to module 33 which produces support data in the SLI format The SLI data representative of the supports identified with numeral 34 is then converged with the SLI data representative of the object, identified with numeral 32, in module 35 The result is PFF file 36, representative of the object and support boundaπes The PFF file is then "hatched" in module 37 in accordance with the style determined by style file 38 using the hatching techniques descnbed in the aforementioned '730 application The intersections between the hatch lines and the object and support boundanes are then used to prepare RLE file 39 A problem with this embodiment is speed As illustrated in Figures 8a-8b the process involves intersecting an STL file 46 with slicing planes, such as that identified with numeral 47 in Figure 8a, to produce segment lists for each cross-section, such as that identified with numeral 48 in Figure 8b The segments are then ordered, intemal segments are removed, and the appropπate end points are joined together to form polygons In Figure 9, for example, the segments 48 are processed in the manner descnbed to form polygon 49 The process is time-consuming because of the number of compaπsons that must be performed to order the segments, and because of the time required to perform Boolean operations on the polygons For a list of N segments, for example, the ordering step requires N 2 compaπsons Moreover, the process of performing a Boolean operation on a polygon compπsing N segments also requires N 2 operations. For both these reasons, the process of forming the build data can be prohibitively long, typically several hours However, one advantage to this approach is that since boundary segments are ordered into polylists, drop width compensation can be performed on these boundaries in a manner analogous to the compensation routines taught in the 730 application A second preferred embodiment designed to overcome these problems is illustrated in Figure 10 As shown, STL file 40 is first compressed through module 41 into CTL file 42 The process of compressing an STL file into a CTL file is descnbed in the aforementioned U S Patent Application Serial No 08/428,951 Second, based on Style infoπnation 43 provided as an input, in module 44, the CTL file is sliced in a manner similar to that descnbed in the '730 application except that only hatch or skin type data is output into an RLE (i e run length encoded) file First, as illustrated in Figure 11a, the triangles making up the STL file are sorted top-down in the z-direction Specifically, as indicated by identifying numeral 50, the triangles are sorted in descending order of the maximum z value for each triangle As shown, the order of the tnangles is A, B, C, D The top-down sort required should be contrasted with a bottom-up sort, such as the one indicated by identifying numeral 51 in Figure 11a, in which the triangles are sorted in ascending order of the minimum z-values of the triangles As indicated, the order which results is: B, C, A, D For each slicing level, a list of active tnangles is then determined through use of a current level indicator, and an index pointer An index pointer is advanced through the list of tnangles for a given level, and any triangles completely above the current level are eliminated from consideration If a triangle is intersected by the current level indicator, it is added to the list The process continues until the index pointer points to a tnangle completely below the current level At this point, the list of active triangles for the level is complete The level indicator is then changed to indicate the next lower level, and the process continues Figure 11b illustrates the process when the current level indicator is at level 52a. The index pointer 53 is advanced from left to right, and the two triangles intersected by the current level, identified in the figure with reference numeral 54a, are added to the list of active tnangles The process then continues until the index pointer is pointing at tnangle 55a. Since that tnangle is completely below the current level, the process stops with the index pointer 53 pointing at triangle 55a Figure 11c illustrates the process when the level indicator is advanced to level 52b. The index pointer is reset to zero, and then advanced from left to right Each tnangle above the level is ignored, and each tnangle intersected by the level is added to the list of active tnangles. In the figure, these tnangles are indicated with identifying numeral 54b The process completes when the index pointer is pointing at triangle 55b since that is the first tnangle encountered which is completely below the level The active tnangles for each sticing level are intersected with that level to form a set of segments in the x-y plane. Since the tnangles bound solid and are oπented so as to face away from the solid region (as explained in U.S. Patent Nos. 5,059,359; 5,137,662; 5,321,622; and 5,345,391 which are hereby incoφorated by reference), the resulting segments also have oπentation. From these segments, without the need of ordering them into boundary loops, the .RLE data descπptive of the object cross-sections can be obtained by using the same hatching algoπthms as described in the '730 application. Figure 12a illustrates a polygonal representation of a cross-section (segments ordered to form boundary loops), while Figure 12b illustrates an .RLE (run-length encoded) representation of the same cross-section. To produce the data, the polygonal representation is overlaid with a plurality of raster or pixel scan lines, and then, a list of start/stop pairs is generated at the points where the raster or pixel lines intersect the polygonal representation, with each point of intersection being associated with an on/off indicator For a given scan line, the on/off indicator for the points of intersection are alternated between on and off status to indicate whether the scan line is entering or exiting a solid. In Figure 12b, for example, the "on" portions of the successive scan lines are identified with numerals 56(1 ), 56(2), 56(3) and 56(11). The .RLE format should be contrasted with the pixel format illustrated in Figure 12c, in which each point inside the solid is represented by a separate data point The problem with this form of data representation is size. At 300 DPI (dots per inch), for example, a 10 inch cross-section requires 9 million bits of information. The process of generating the RLE data for the object cross-sections is illustrated in Figure 13a-13c. As shown in Figure 13a, for each cross section, such as the cross-section identified with numeral 57 in the figure, an array of lists, identified with numeral 58, is created, in which each list in the array corresponds to a scan line extending at a given y-level in the x-direction. Then, consideπng in turn each segment in the cross-section, the intersections between each segment and the scan lines are noted, and data representative of these intersections is added to the respective lists in the array. Figure 13b, for example, illustrates the additions, identified in the figure with numeral 59, to the lists through consideration of segment one. The specific data items added to the list for each "y" location contain two pieces of information: a quantitative volume (QV) value, and the x-location of the intersection. An intersectaon at which the segment is increasing in the y-directaon has a QV of 2. An intersection at which the segment is decreasing in the y-direction has a QV of -2. If the segment oπginates or terminates at a scan line, the intersection counts as a "half-hit", i.e., the associated QV is either 1 or -1 depending on whether the segment is increasing or decreasing in the y-direction. In Figure 13b, for example, segment 1 is increasing in the y-direction. Thus, the QV values associated with the intersection of this segment with successive scan Iines are respectively 1, 2, 2, 2, and 2 (assuming that the scan line does not meet the tip of segment 1). Moreover, the x-locabons of the intersections between segment 1 and the successive scan Iines are respectively 126, 124, 122, 120, and 118. As shown, the data added to the array incoφorates these values. Figure 13c illustrates the additions to the array through consideration of segment 2 That segment increases in the y-direction, and originates and terminates at two successive scan lines The x-location of the intersection of the first scan line is 144 while for the second, is 126 The two additions to the array which incoφorate these values are identified with numerals 60(1) and 60(2) The purpose of the half hits can be understood through consideration of Figure 14 As shown, each scan line is associated with a running QV total which is updated every time the scan line crosses a segment using the QV value associated with the point of intersection with the segment If the scan line is inside solid, the running QV value is 2 while if it is outside solid, the QV value is 0 Thus, when a scan line is outside solid, and crosses a boundary, the necessary implication is that the scan line is now inside solid The running QV total should then be updated with a value of 2 to indicate that it is now inside solid Conversely if the scan line is inside solid and crosses a boundary the necessary implication is that the scan line is now outside solid or has entered a second solid object which is overlapping the first object A value of -2 or 2 should then be added to the running total to indicate the transition If the scan line crosses a vertex such as indicated at point A in Figure 14 the scan line actually intersects two segments when entering solid Each segment should thus only contπbute a value of 1 to the running QV total That is why the QV value associated with these vertices is kept to either 1 or -1 It should be noted that it is possible for a scan line to cross a vertex without the running QV value changing state As illustrated by point B of Figure 14 the segments forming the vertex have respectively QV values of -1 and 1 at the point of intersection The result is that the running QV total associated with the scan line is unchanged Additional information about quantative volume (QV) can be found in the previously referenced '730 U S patent application After the scan-line intersections for all the segments have been added to the list, the list for each scan line is then sorted in ascending x-order A Boolean extraction routine is then applied to extract the correct Booleanized segments for each scan line The preferred extraction routine involves maintaining a running QV count in which the QV value of each successive data point in the list is added to the running total Any data point which has a QV of 2, i e , a "start" point, when the running QV total is 0 (i e transitions from 0 to 2), and any data point which has a QV of -2, i e , a "stop" point, when the running total is 2 (i e transitions from 2 to 0), is kept The process is illustrated in Figure 15 in which successive steps thereof are identified with numerals 61, 62, 63, 64, 65, 66, and 67 A current item pointer, identified with numeral 68, is used to point to successive items in the original list A "Kept" list, identified in the figure with numeral 70, is also used to retain the start and stop points which meet the prescπbed conditions described above As shown, through this process, only the first start point, i e , (start 20), and the last stop point, i e , (stop 89), are retained The result is the RLE data descπptive of a line of a cross-section of the object Applying the technique to all lines for all cross-sections results in an RLE descπption for the object It should be appreciated that it is not necessary to sort the segments, formed by intersecting the triangles with the slicing planes, into polyhsts (as descnbed in the '622 patent) in order to form" polygonal representations of the object cross-sections As discussed, the sorting of segments into polyhsts is a time-consuming operation Moreover, it should aiso be appreciated that the RLE data formed is successfully unioned even when the STL file has not been properly unioned or separated (i e the STL file contains overlapping object elements) A benefit of the RLE representation over the polygonal representation is that Boolean operations are much simpler and faster The Boolean extraction algoπthm has already been discussed Several others are Boolean addition subtraction, and intersection operations To perform these operations most efficiently, it is advantageous to express the RLE data in absolute terms as opposed to relatave terms For example, a line starting at x-position 100 and staying on for 30 pixels should be represented in terms of a pair of start/stop points in which the start is at position 100, and the stop is at position 130 Thus, with reference to Figure 16, the RLE data for line A identified with numeral 71 in the figure and that for line B, identified with numeral 72 in the figure, are represented as follows A = [(start 20), (stop 48), (start 60) (stop 89) ], B = {(start 37) (stop 78)] Computing the Boolean addition of these two lines involves merging the two sets of data, while keeping the merged list sorted in the x-direction The result is [ (start 20), (start 37), (stop 48), (start 60), (stop 78), (stop 89) ] The merged list is then subjected to the Boolean extraction algoπthm discussed earlier wherein for example the start locations are assigned QV values of 2 and stop locations are assigned QV values of -2 and only those locations resulting in QV transitions from 0 to 2 (start) or from 2 to 0 (stop) are kept The result is the data pair [ (start 20), (stop 89) ], representing the Boolean addition A + B, which is identified with numeral 73 in Figure 16 To compute the Boolean subtraction of two lines involves the identical steps discussed above in relation to the Boolean addition operation except that before the two lists are merged, the signs of the QV values of the list which is being subtracted is reversed such that start transitions become stop transitions and vice-a-versa The result of the operation A - B is identified in Figure 16 with numeral 74 To compute the Boolean intersection of two lines involves the identical steps as the addition operation except that the extraction routine is performed starting with an initial QV value of -2 The intersection between A and B is identified in Figure 16 with numeral 75 Two-dimensional Boolean operations can also be easily performed For two-dimensional areas, each represented by a plurality of RLE lines expressed preferrably in absolute terms, Boolean operations are performed by performing successive Boolean line operations on each successive pair of corresponding lines in the respective areas Figure 17 illustrates the process The set of lines identified with numeral 76 represents area A, while the set of lines identified with numeral 77 represents area B The Boolean addition, A + B, of these two areas is identified with numeral 78, while the Boolean subtraction of these two areas, A - B, is identified with numeral 79 A drawback of using RLE data in relation to polygonal data, on the other hand, is the amount of memory required To store every layer in RLE form at high resolution might require over 100 MB of storage for a typical part This is too large for mam memory, and even having to store such large files on disk is problematic. The problem is compounded by the divergence between the order of part building, which proceeds from the bottom up, and the order of constructing support structures, which, as descnbed herein after, proceeds from the top down. As discussed hereinafter, an output file is required for constructing supports in which, for each cross-sectaon, an .RLE descπption is provided for that cross-section, as well as the Boolean summation of every cross-sectaon above the current cross-section. Basically, the technique involves computing the Boolean subtraction between the .RLE description of a cross-section and the RLE representataon of the "current total" for that cross-sectaon, i.e., the Boolean union of all layers above the current layer. Pseudo-code for this basic technique is shown in Figure 18, in which get_part (level) refers to a function which provides the .RLE representataon of the cross-sectaon at the prescribed level, boolean_subtract (curτent_total = area A, part_for_layer = area B) refers to a function which provides the result of Boolean subtracting area A from area B, and boolean_add (area A, area B) refers to a function which provides the Boolean addition between area A and area B An algoπthm for performing memory management which permits supports to be constructed without requiring the entirety of the part and current total data to be simultaneously stored in memory will now be described. The preferred algorithm proceeds in two stages. In the first stage, the layers of the part are successively considered starting from the top of the part while maintaining a running total of the Boolean summation of the layers of the part Upon encountering a layer, the current total for the layer (i e updated running total) is computed by calculating the Boolean addition between the area of the running total from the previous layer and the area of the current layer. However, instead of storing the current total data for all the layers, only the current total data for intermediate layers, i.e., every Nth layer where N might be 100, are stored. The rest of the current total data is discarded This first stage is illustrated in Figure 19 in relation to part 80 and the associated supports, identified in the figure with numeral 81 The top-down generation of the current totals for the respective layers is identified with numeral 82, and the intermediates of these are identified with numeral 83. Pseudo-code for this first stage is illustrated in Figure 20 in which the get_part function is that described earlier in relation to Figure 18, and the boolean_addιtιon function is that described earlier in the discussion of Boolean operations The second stage involves selecting an intermediate layer and performing a top-down computation, in the manner descnbed previously, of the current totals for all the layers between that intermediate layer and the next intermediate layer. The data, consisting of the part and current total data for each layer, is then output from the bottom-up. When this has been accomplished, the current intermediate layer, and the data between it and the next lower intermediate layer can be deleted, and the process repeated for the next higher intermediate layer This second stage is illustrated in Figure 21 , in which compared to Figure 19, like elements are referenced with like identifying numerals. Four steps, identified with numerals 84-87, of this second stage are shown. In step 84, the current totals for all the layers between intermediate layers 14 and 15 (e.g. the bottom of the part or object), identified in the figure with numeral 88. are determined and stored. Next, in step 85 the supports for these layers are determined using the methods described hereinafter, and then output. The part and total data between 14 and 15 is then deleted Then, in step 86, the part and total data for each layer between 13 and 14, identified with numeral 89 in the figure, is determined and stored Finally, in step 87, the supports for these layers are determined and output for building The data for these layers is then deleted The process then repeats itself for every intermediate layer It should be appreciated that this algorithm drastically reduces the memory requirements for the support generation process If N is the number of layers between two successive intermediate layers, then the number of layers which is stored at a time is equal to the number of intermediate layers plus 2N (since part and total is required) If T is the total number of layers, the number of stored layers is equal to T/N + 2N Optimal memory usage is then obtained when N = the square root of (T/2) Thus, for a total of 5000 layers, the optimal number of intermediate layers N is 50 The total number of layers that must be stored at any time is thus 200 Memory requirements can be reduced further by extending the aforementioned algorithm to two levels of intermediate layers As shown in Figure 22. the algorithm proceeds in three stages, depicted in the figure with identifying numerals 90, 91, and 92 In the first stage, identified with numeral 90, the first level of intermediate layers is determined In the second stage, depicted with numeral 91 , a second level of intermediates is determined between two of the first level of intermediates Then, in stage three, depicted with numeral 92 in the figure, the current totals for all the layers between two successive second level intermediates is determined and stored After computing supports for these layers, the data is discarded, and the process is repeated for the next second level intermediate When all the second intermediates associated with the current first level intermediate have been processed, the next first level intermediate is processed If the number of first level intermediates is N, and the number of second level intermediates is M, then the memory requirements for this three-stage process is (T/N) + (N/M) + 2M If T = 5000, N = 288. and M =14, then the number of layers that must be stored at a time is 66. Since this three-stage process increases computation time, the two-stage process is preferred unless very thin layers or large numbers of layers are involved, in which case the three-stage process may be preferable As discussed, the RLE data for a given layer consists of a set of start and stop transitions, with an x-location associated with each transition. The data depicted in Figure 23, for example, corresponds to the following start and stop locations and raster lines: raster line A = [ (start 20), (stop 48), (start 60), (stop 89) ], indicated by reference numerals 102, 104, 106 and 108 respectively, and raster line B = [ (start 35), (stop 72) ], indicated by reference numerals 112 and 114 One method for stonng this data consists of a linked list of start/stop transitions, such as is depicted in the pseudo¬ code of Figure 24 Compared to an aπay, a linked list is preferred because it easily allows for flexibility and variability in the number of transitions required per scan line The problem is that it results in the use of large numbers of dynamically allocated small memory chunks which can significantly degrade performance for at least three distinct reasons First, dynamic memory allocation is tame consuming since it requires systems calls. Second, each chunk of dynamic memory has a hidden storage overhead associated with it which is used for book-keeping Third, logically-adjacent pieces of information are located in non-contiguous memory leading to a large number of cache misses. To overcome these problems another form of data structure is more preferred At a resolution of 1200 DPI, a transition in a typical part can be represented with 15 bits Thus, a 32-bιt word (with two spare bits) can be used to represent a start/stop pair This data structure is depicted in psuedo code of Figure 25 The "last" flag is utilized to indicate whether the start/stop pair is the last in the set for a particular scan line If so, the "last" bit is set to a logical "1 " If not, the bit is set to a logical "0 " In this case, the next start/stop pair in the sequence is stored in the immediately adjacent memory location This scheme enables large numbers of transition points to be stored in contiguous blocks of memory, with 2 bytes provided per transition An example of this scheme is provided in Figure 26 wherein like elements are identified with like numerals as used in Figure 23 As shown, line A consists of two transition pairs [ (start 20), (stop 48) ] and [ (start 60), (stop 89) ], elements 102, 104 106 and 108 respectively which are stored in contiguous 32-bιt words as shown The "last" bit 122 in the first word is reset to a logical "0" to indicate that additional data for the scan line follows, while the "last" bit 124 for the second word is set to a logical "1" to indicate that no additional data follows Line B consists of only a single pair of start/stop locations as indicated [(start 37), (stop 78)], refered to with numerals 112 and 114 respectively, and wherein the last bit 126 is set to logical 1 so as to indicate that no additional data for line B follows Reference numerals 132, 134 and 136 refer to other used bits associated with each 32-bιt word The .RLE data is not created initially in the above described packed format Instead, as discussed in relation to Figure 13a-13c, it is initially created in an unpacked format, and then converted to the packed format In summary, a memory block is allocated for stonng transitions Pointers are used to indicate where data associated with each raster line starts ("current raster line pointer or "current list" pointer) and a pointer indicating where unallocated memory begins ("next location available or "next free locataon pointer) Each four byte (32 bit) word in this memory block is defined such that the first 15 bits are used to store the x-location of the transition, and the second 15 bits are used to store the qv of the transition The 31st bit is used to define a "used" flag which indicates whether the word has been allocated and used. The 32nd bit is used to define an end flag which indicates whether the entry in that word is or is not the last transition entry for given scanning line for which the word is associated. Initially each raster line may be allocated one or more words for stonng data As transitions for each boundary segment are entered into the memory block, they are added to the lists associated with the raster lines from which they are derived In adding each new transition point to the raster line lists, several situations can be encountered First, If there is no transition data in the memory block associated with a a given raster line, the transition data is added to the word associated with the "current list pointer" for that raster line Second, if transition data exists at the word assoicated the current list pointer for the given raster line, the word following (i e "following word") the last recorded transition point for that raster line (i.e for that current list pointer) is checked to see if it has been used If not used the new transition data is entered there Third, if the "following word" is occupied, then the word before the current list pointer (i e "pπor word") is checked to see if it is being used If not, the current list pointer and all recorded transition data (for the raster line) is shifted by one word and the new transition point data is added to the end of the shifted list Fourth, if the "prior word" is occupied all the transition data for the raster line (including the current list pointer for that line) are moved to the word marked by the "next location available" pointer, the new transition data added, the oπgmal word locations of the transitions marked as being available for adding new data, and the next location available" pointer moved to the locataon following the just moved words and added word Vaπous modifications to the above outlined procedure can be made For example, different sized words can be used, bit allocations can be varied, initial allocation amounts for each raster line can be varied, initial allocations for each raster line can be avoided and memory locations allocated as additional raster lines are needed to completely process the input segments, additional steps can be added to better control memory use and the like The above decπbed process is exemplified in the description to follow and associated figures Figures 27a & b are based on the same data found in Figure 13 and as such like elements are referenced with like identifying numerals illustrate the process A large area of memory 93 is allocated to hold the RLE transitions, and pointer 101 is used to indicate the next available memory word (32 bits) In this example, the word format includes the following bit designation the first 15 bits 142 record the valueused to store the x-location of the transition, the second 15 bits 144 record the value of the qv of the transition The 31st bit 146 is the "used" flag which indicates whether the word has been allocated and used The 32nd bit 148 is last flag or "end" flag which indicates whether or not the transition is the last recorded transition for the raster line Figure 27a depicts the situation before any transition data is added to memory 93 For procedural reasons as will be made clear hereinafter, the first word in area 93 as shown, is marked as used The "next free locataon" pointer 101 points to the second word in the area Next, an array 58 of pointers is set up with all the pointers initialized with their "used" bits set to zero As discussed above, each pointer is associated with a scan line and is used to locate the memory locataon for the first word (i e for the first transition) associated with that scan line This pointer is called the ' current list" pointer since it points to the first word in the list of transitions associated with the current scan line being considered To add a transition for a particular scan line to the array, if the pointer in the array is on a word with a "used" bit set to logical 0, the location of the pointer is considered to be free and the transition is allocated to that word of memory Figure 27b depicts the situation wherein a first transition has been entered into memory for five scan lines The process of adding a transition for a scan line that has a non-zero ' used" flag in the position of the "current list" pointer 94 is illustrated in Figure 28a & b Figure 28a depicts two words 150 and 160 which are already entered as belonging to the scan line associated with current lister pointer 94 Word 150 includes bit allocations 150, 154, 156, and 158 having the same definitions assocated with bits 142, 144 146 and 148 of Figure 27b Similarly word 160 includes bit allocations 162, 164, 166, and 168 Elements 156 and 166 give the value of the "used' flag Elements 158 and 168 indicate whether or not the word (i e transition) is the last transmission thus far recorded in the current list As can be seen element 158 indicates that word 150 is not the last word, whereas 168 indicates that 160 is the last used word in the current list First the "used" flag in the next word 170 after the end of the current transition list which flag is identified with numeral 96 in Figure 28a, is checked to see if the word is available If the "used" flag is set to logical 0 the word is available for stonng new transition details If it is set to logical 1 the word is not available If available, as shown in Figure 28a, then the new transition details can be placed into this word The current list as modified by the addition of a new transition is depicted in Figure 28b In figure 28b the new transition details 97 are added to word 170, the value of 'end' flag element 168 is changed from "1" to "0", and end flag element 178 of word 170 is given the value "1" as 170 is now the end word of the current list If the next word after the end of the current transition list is not available, then the availability of immediately preceding word before the beginning of the current transition list is checked This checking occurs by evaluating the value of the used" flag of this immediately preceding word If available (as indicated by a 0" value) then the entire list is shuffled back one word and the new transition is placed in the word which has just been cleared This process is illustrated in Figures 29a- 29b, in which, compared to Figures 28a-28b like elements are referenced with like identifying numerals As shown in Figure 29a, the current list' pointer is associated with word 150, the list ends with word 160 and the next word after the end of the current list, identified in the figure with numeral 170, is unavailable (due to the value "1" in element 176) while the word just before the beginning of the list, identified with numeral 180, is available (due to the value "0" in element 186) The consequences of these evaluations are shown in Figure 29b, wherein the transition values previously associated with words 150 and 160 are shifted to be associated with words 180 and 150 respectively The "current list" pointer is also shifted to word 180 and the new transition infoπnation is added to now available word 160 As a further result the end" flag remains associated with word 160 though it is no longer associated with the transition at x-value 60 (previous element 162 new element 152) but instead is associated with the transition at x-value 12 (previous element 172 new element 162) In other words, the entire current list is shuffled back by one word and the new transition 97 ts stored in the cleared locataon If there is no room in front of or behind the current transition list (i e the word immediately preceding the current list pointer and word immediately following the word containing the true end of list flag), the entire current list is copied into the space beginning with the word indicated by the "next available location" pointer, and the new transition is added to the end of this copied list The "used" flags of the original memory words in which the list was stored are then reset to indicate that these onginal memory words are now available for use by scan line lists immediately preceding and i mediately following these original locations This process is illustrated in Figures 30a-30b. in which, relatave to Figures 28a-28b, 29a-29b, like elements are referenced with like reference numerals Figure 30a illustrates that the word 170 after the end 160 of the current list, as well as the word 180 preceding the word 150 containing the current list pointer are both unavailable due to "used" flags 176 and 186 be set to "1" Figure 30a further illustrates the word 200 where the next availiable locataon" pointer is found Word 200 follows the already entered transition points for all scan lines Consequently, no new transitions for the instant scan line can be entered in consective memory locations to those locations 150 and 160 already containing transitions associated with the scan lines As illustrated in Figure 30b , the entire current list (transitions originally located in words 150 and 160) is copied into the area beginning with the word 200 pointed to by the next free locataon pointer 101 The "used" flags in the old memory, identified with numeral 100 in Figure 30b. are reset to indicate that this memory is now available The current list pointer 94, is updated to point to word 200 , the new transition 97 is added to the end of the list at word 220 The "next available locataon" pointer, identified with numeral 101 is then updated to point to the word 230 immediately following the word 220 containing the last entered transition 97 (i e the end of the list) Of course, if desired one or more empty words can be left between the last entered transition 97 at word 220 and word pointed to by the "next available location pointer This scheme is particularly efficient due to the nature of RLE data Because the data is used to describe solid geometrical objects the number of transitions on a particular scan line is usually the same as the number of transitions on a neighboring line This property is illustrated in Figure 31 An object cross-section is depicted from the top wherein spaced raster or scan lines are shown At the right of each scan line the number of transitions associated with that scan line is shown Thus, if it is desired to add a transition to a particular scan line, it is likely that a transition will be added to a neighboπng scan line When an area of memory gets freed, as described in Figures 30a-30b and the accompanying text, it is likely that neighboring list will have transitions that canbe stored in this area, as illustrated in Figures 28a-28b, and 29a-29b, and the accompanying text Thus, large arrays of memory develop fewer gaps than would occur with random data Also, there will be fewer misses from the data stored in cache When all the segments have been processed, the resulting lists are then sorted in the x- direction The correctly booleanized lines are then extracted in the manner previously described, and the extracted lines are stored in the packed format previously described This embodiment operates directly on an STL file without requiπng rounding of vertices to slicing planes, and thus avoids at least some quantization error Some vertical and horizontal quantization error is introduced, however, through the generation of RLE data since slicing planes will only be located at discrete levels in the vertical direction and since honzontal transitions will be limited to pixel boundaries An example of these issues is depicted in Figure 32 which represents quantization decisions associated with representing the on/off transition points 322 324, 326, 328, 330, 332, and 334 for raster lines 302, 304, 306, 308, 310 312. and 314 The center line of each raster line is depicted by respective dashed lines assoicated with boundary segment 300 which crosses through a plurality of pixels In the figure, the region to the πght of the line is considered to be within the object and the region to the left is considered to be outside the object For each raster line only a single transition pixel can selected to represent the edge of the object regardless of how many pixels on that line are intersected by the boundary Though there are many ways to determine which pixels will form the boundary of the object, the depicted approach selects the boundary pixel for a given raster line as the pixel which contains both the line segment and the center line of the raster line. In the event that the center line of the raster line meets at exactly the boundary between two pixels, a decision is made as to whether or not to emphasize object (i.e. solid) or non-object (i.e. hollow). As depicted for raster lines 302, 306, 310 and 314 the decision to emphasize hollow was made. A number of transition selection alternatives exist For example, one may elect to emphasize solid by selecting the transition to occur such that any pixel through which the line passes is counted as part of the object. Oppositely, one may elect to emphasize hollow by selecting the transition to occur such that only those pixels which are completely within the object boundary are included as part of the solid region As an intermediate alternative, one may take an average of the transitions from the previous two alternatives Other schemes for determining transition locations may involve determinations of area percentages of the solid or hollow for boundary region pixels, and the like Implementation of some of these techniques may be aided by use of the techniques described in previously referenced patents and applications expecially those involving slicing techniques As a final example, an alternative may involve subdividing a pixel and basing a decision based whether the segment intersects one or more of the subpixels Whatever approach is used, however, consistency is desired in the approach used in relation to both the part and supports Data Compensation Techniques: Compensation is easily performed by moving the endpoints of the transitions in or out keeping in mind that endpoints from adjacent segments should not cross To avoid having a support touch a part, for example, the .RLE date for the part could be expanded, and then Boolean subtracted from the current total data to get the .RLE data descπptive of the support region Alternatively, the current total data could be expanded, and the support data computed as the Boolean difference between the expanded current total data and the part data. Or, the support data could be computed as the Boolean difference between the current total data and the part data Then, the support data is expanded The actual support data is then computed as the Boolean difference between the expanded support data and the original part data. Compensation to adjust for drop size along the scan direction is easily performed as long as the DPI is at a higher resolution than the drop diameter. Compensation in the y-direction is more difficult, but can also be accomplished by stepping in smaller increments than 300 DPI It is useful to be able to convert RLE data into vector data. As shown in Figure 33, the technique involves connecting two consecutive "on" points or consecutive "off points to form vectors unless there is an intermediate point between the two, in which case the connection is impermissible. In Figure 33, for example, it is permissible to connect point a and point a' but it is impermissible to connect point a to point c. The reason is that point b is between the two. Support Data Generation: A preferred process of creating data for support structures will now be descnbed. The process begins with data provided from the above descnbed data manipulation techniques As described above, the Data Manipulation Subsystem provides object (i e. part) data and "total" data for each layer The part data for a given layer is a senes of start and stop points in adjacent raster lines which define the XY locations of the part at that layer The "total" data for a given layer is a series of start and stop points in adjacent raster lines which define the Boolean union between the XY locations of the part at that layer and any desired support at that layer Such data is illustrated in Figures 34a-34c Figure 34a illustrates the part data P[1] to P[10] for each layer (i e cross-sections, lamina) 1 to 10, respectively, for a "peanut" shaped part shown floating in the z-x plane In Figure 34a only a single RLE line for each of cross-sections P[1] to P[10] is shown The start transitions are identified with the 'V" symbol while the stop transitions are identified with the "V symbol As can be seen the part data tracks the boundary (i e extents) of the part Figure 34b illustrates the "total" data T[1] to T[10] for each layer 1 to 10 respectively, for the part It is also defined in terms of start and stop transitions However, unlike the part data, it does not necessarily track the boundary of the part As discussed above the "total" data for a given layer is the Boolean union of the part data for all the layers above the given layer Figure 34c illustrates a cross-sectional view (in the X-Y plane) of both the part and total data for a given layer This data, identified as P[ι] and T[ι], respectively, comprises a plurality of start and stop transitions which are arrayed along hash lines H[ι] in the X-Y plane In a preferred embodiment the hash lines would be oriented parallel to the x-axis However, as indicated other oπentations of hash lines are possible In a preferred embodiment the combined object and total data is used to determine the start and stop transitions for the supports, one layer at a time If a single type of support is to be used in all regions requiring supports, a single support style can be defined which can be applied to each layer in the region defined as the difference between the total data for a layer and the part data for that layer On the other hand, as discussed in U S Patent Application No 08/534,813 , it may be advantageous to use different types of support structures for different locations depending on how close or far away any up-facing and/or down-facing surfaces of the object are Furthermore, it may be advantageous to use different support styles depending on how far the region is from object boundaries on the same layer Techniques for performing horizontal compaπsons are described in above referenced U.S Patent Application No 08/428,951 which are applicable to the current invention to aid in defining support regions For example, it may be advantageous to utilize two different support styles one for use when a region is a few layers below a down-facing surface and one for use elsewhere Alternatively, two physical support styles can be used in combination with a third "no-support" style wherein the no-support style might be applied to the region that is within 1 or 2 pixels of the boundary regions of the part or wherein the part surface above the object as a normal to the vertical which is greater than some cntical angle Many additional embodiments utilizing multiple support styles are possible and they can be readily implemented by the teachings herein and those incoφorated by reference (particularly U S Patent Application Nos 08/475,730; 08/480,670, 08/428,951, and 08/428.950) Additionally, the teachings herein can be applied to what might be termed inteπor object supports, wherein single or multiple supports styles may De used in the process of forming inteπor portions of objects. Examples of such techniques, as applied in stereolithography forthe purpose of making investment casting patterns are described in U.S. Patent Application Serial No. 08/428,950 previously incorporated To further explain how one might define data for different support regions the following example is given which corresponds to the hybrid support example descnbed in U.S Patent Application No. 08/534,813 In terms of this example, three categories of supports are recognized: (1 )thιn, fiber-like columns spaced in a checkerboard pattern; (2) more substantial 3 x 3 pixel columnar supports; and (3) intermediate or transitional layers Assuming layer "n" is about to be built, the technique involves determining how close each portion of layer "n" is to an up-facing and/or a down-facing surface of the object. In the present embodiment, if a portion of layer "n" is within "r" layers (e.g. 5-10 layers) of a down-facing surface or within "u" layers (e.g 5-10 layers) of an up-facing surface, the checkerboard category of supports is to be built for that portion, if between s" (s=r+1 )and "t" layers from a down-facing surface (e.g. 6-10 or 11-15 layers) and more than "u" layers (e g 5-10 layers) from an up-facing surface, the intermediate or bridge category of supports is to be built, and if more than "u" layers (e.g. 5-10 layers) from an up- facing surface and more than "t" layers (e g 10-15 layers) from a down-facing surface, the 3 x 3 columnar support is to be built The above example is illustrated in Figures 46a and 46b which depict identical side views of an object with a gap between an up-facing surface and a down-facing surface of an object. Figure 46a depicts the side view along with hypothetical levels and regions upon which formation of different support structures will be based. Figure 46b depicts the side view wherein the gap is filled with various types of support structures according to the layout of the hypothetical levels and regions of Figure 46a. More specifically, Figure 46a depicts a down-facing object surface 402 ana an up-facing object surface 400 which are separated by a spacing compπsing the regions 404 410, 408 and 406 Region 404 is located within "u" layers of up-facing surface 400 and region 406 is located with "r" layers of down-facing surface 402. Region 408 is located between 'r" and "t" layers from down-facing surface 402 and is simultaneously located more than "u" layers from up-facing surface 400 Region 410 is located simultaneously more than "u" layers from up-facing surface 400 and more than "t" layers from down-facing surface 402. Region 404 and 406 are to be formed with checkerboard type supports, region 408 is to be formed with transition type supports (e.g. completely solidified) and region 410 is to be formed with 3 by 3 column supports. Layers 414, 412. 424, and 416 are shown to be completely within regions 404, 406, 408. and 410 respectively Therefore these layers will be formed with a single type of support structure over their entire area. On the other hand layers 418 , 420, and 422 are shown to be partially located within regions 404 and 410, 410 and 408, and 408 and 406, respectively. Therefore, these layers will be formed with different types of support structure dependent upon XY location of each portion of the layers. Figure 46b depicts solid object regions 432 and 430, above and below down-facing surface 402 and up-facing surface 400, respectively The region 404 and 406 are indicated as being filled in with checker board (one pixel on, one pixel off) supports. Region 410 is indicated as being filled by 3 by 3 column supports (3 pixels on, one pixel off) supports. Region 408 is indicated as being filled in by a solid region of supports. This embodiment may be presented in equation form. In presenting these equations, the following terminology is used' C n (D) ' the area elements of layer n over which the "checkerboard" category of supports should be built as determined from down-facing surfaces. C n (U) the area elements of layer n over which the "checkerboard" category of supports should be built as determined from up-facing surfaces. B n (D) ' the area elements of layer n over which the "bridge" category of supports should be built as determined from down-facing surfaces. S n the area elements of layer n over which the 3 x 3 pixel column category of supports should be built P the area elements of the part at cross-section "I" P n the area elements of the part at cross-section "n" T n the area elements of the total data at cross-section "n" ∑ Boolean summation of area elements + Boolean union of area elements Boolean difference of area elements. n. Boolean intersection of area elements. r the number of layers below a down facing feature which are formed with checker board supports. u the number of layers above an up-facing feature which are formed with checker board supports s r+1 = the number of layers below a down-facing surface at which transition- type supports end. t' the number of layers below a down-facing surface at which transition-type supports begin With this terminology in mind, the following equations define the preferred method of determining supports for layer "n" according to this embodiment: (1 ) C (D) = ∑ P - P n i -n * \ (2) C n (U) = ∑ (/ » , - P n ) π T π (3) B n (D) = ∑ P C n (D) -P n -C n (U) (4) r - P - C (D) - C (U) - B (D) Equation (1 ) indicates that the area of layer "n" over which the checkeΦoard category of supports should be built, as determined from down-facing surfaces is calculated by taking the Boolean union of the part data of the r' layers above layer n and then calculating the Boolean difference between the data representing this unioned area, and the part data for layer "n" Equation (2) indicates that the area of layer "n" over which the checkeΦoard category of supports should be built, as determined from up-facing surfaces, is calculated by taking the Boolean union of the part data of the "u' layers below layer "n", calculating the Boolean difference between the data representing this unioned area, and the part data for layer "n ", and then calculating the intersection between this data and the total data for layer "n" The purpose of this last calculation is to avoid building supports when in fact there are no part layers above layer "n" Equation (3) indicates that the area on layer "n" over which the bπdge supports should be built, as determined from down-facing surfaces, is calculated by 1 ) taking the Boolean summation of the part data from layers "s" through "t" above layer' n", and 2) then differencing from the summed date of step 1 ,t the data representative of the areas over which the checkerboard supports will be built on layer n (below down-facing and above up-facing surfaces) and data representative of the areas over which the part itself will be built on layer "n" Essentially, this equation establishes a priority between bridge and checkerboard supports. It requires that in areas which are both within "u" layers of an up- facing surface, and within "s" to "t" layers of a down-facing surface (such as an area below a continuously curved surface), that priority will be given to the building of checkeΦoard supports Finally, equation (4) provides that the area on layer "n" over which the 3 x 3 pixel columnar supports are to be built is determined by taking the total data for layer "n" and determining the Boolean difference between this data and 1 ) the part data for layer "n", 2) the data representative of the area or areas of layer "n" over which the checkerboard supports are to be built, and 3) the data representative of the area or areas of layer "n" over which the bπdge supports are to be built. As is apparent from the above discussion, equations can be defined for various regions where different types of support structures may want to be formed. Figure 37 depicts an arch type support structure which requires a different build pattern as one gets progressively closer to a down facing surface 24 As indicated, the arch-type support starts at surface 23 which may be the surface of a build platform, an up-facing surface of the object, or a surface assocated with previously formed supports As such this support structure is a hybrid support with many (e g 10 or more) different support styles required for its formation Of course it would be possible to add a number of layers of checkerboard supports between the tops of the arches and the down-facing surface being supported Once this data has been determined the next step in the process is to format the data for output to the control computer As discussed the control computer will load this date as well as object data in the bit map to drive the print head, as well as the X- Y-, and Z-stages Style files are used for this purpose, one for each category of object structure and support structure A Style file for a given object or support type is the core pattern which is repeated throughout the area in which the category of object or support is to be built Style files are used to modulate the build pattern associated with a given region This data modulation technique simplifies data manipulation and storage requirements For example, the Style file associated with the "checkerboard" category of supports in the present embodiment is the 2 x 2 pixel pattern shown in Figure 38a As a second example the Style file associated with the 3 x 3 pixel column supports in the most preferred embodiment is the 4 x 5 pixel pattern shown in Figure 38b Of course many other style patterns are possible These Style patterns are repeated one after another, starting typically at (x,y) location (0,0) so as to define a repetitive pattern in XY space This overall pattem is associated with the corresponding start and stop transitional data for object and support regions The combination of Style file information and object information may occur before transfer of data to the control computer or may occur after transfer Typically object and style information are combined into a single date set after both are transfered to the control computer At present, the preferred Style file associated with the part is simply a 1 x 1 solid pixel pattern, indicating that the interior of the part is always solid At present, the most preferred replication of patterns is fixed in the X-Y plane With regard to the most preferred 3 x 3 support patterns the result is that some of the 3 x 3 pixel columns may get diminished at part boundaries This effect is illustrated in Figure 39a As shown, portions 30 and 31 of the 3 x 3 pixel columns are not built because of their proximity to part boundary 32 The result is that these two supports have diminished surface areas If the columns are not retracted from the part boundary this presents little problem since the formation of the part will form the other portion of each partially formed column However building supports in contact with the part tends to damage object surface finish, thereby resulting in another problem In the event the supports are retracted from the part, a solution to this problem is to allow the pattem of replication to vary to allow the 3 x 3 supports to track the part boundary This approach is illustrated in Figure 39b Gradual changes in support column position can be achieved using offset pixel pattems as descnbed in U S Patent Application 08/534,813 As mentioned above, another problem that sometimes occurs is that the 3 x 3 support columns are sometimes built in direct contact with the part This problem is illustrated in Figure 39c As shown supports 33 have been built in direct contact with part 33 (the supports 34 shown in phantom are below the part and illustrated solely for purposes of completeness). A solution to this problem is to move back these supports by 1 pixel or more to space the supports from the part. This can be accomplished simply by adjusting the start and stop transitional data for the supports In the present embodiment, this adjustment is optional because of the trade-off involved' by backing off the support by one pixel, the surface area of the columns will be diminished, possibly causing an accumulation problem A few caveats are in order about the preferred method of performing the Boolean calculataons As has been discussed, the data involved in these calculations is formatted as a seπes of start and stop transitions It has been discovered that this format facilitates the Boolean calculataons by allowing them to be performed as a series of aπthmetic calculations For example, to perform a Boolean differencing operation between two sets of transitional data, it is only necessary to arithmetically subtract corresponding start and stop transitions from one another The result is a significant improvement in computational speed The reason is that Boolean operations involving N data points based on polygonal data are essentially N 2 operations whereas arithmetic operations using start and stop transitional data are essentially proportional to N Another point is that the intermediate Boolean union data calculated for layer "n", i e , the Boolean union of the part data "r" and "u" layers above and below layer "n", and between "s" and "t" layers above layer "n ", cannot be used in any subsequent processing The reason is the lack of "memory" associated with the Boolean union operation as illustrated by the following equations n 'L (5) ∑ 1 = L (arithmetic) rl 'L (6) ∑ = 1 (Boolean ) As indicated, with the arithmetic operation, the nth item in the summation has an effect on the final sum which can be subtracted out when the calculataons are performed for the next layer. With the Boolean operation, on the other hand, the nth item does not necessarily have any impact Thus, the effect of this item cannot necessarily be subtracted out when the calculations are performed for the next layer. Though equations (1) to (4) above produce exact results, they may lead to excessive computation time. As such, in some circumstances, it may be desireable to utilize equations that may give approximate results but involve fewer computations Excessive calculataons can be avoided by making the assumption that the slope of a part surface does not change sign in a given number of layers (e g. 10 layers, about 10-20 mils) or that any change in direction represents a negligible vaπation in cross-sectional position In other words, the assumption is that the part surface does not change rapidly or drastically. This point is illustrated in Figures 35a-35b. Figure 35a illustrates a part which is consistent with the assumption. As can be seen, the slope of the part surface, identified as S in the figure, does not change sign over or direction over a given number of layers, for example 10 layers. Figure 35b, on the other hand, shows a part which is inconsistent with the assumption that direction of the slope of the surface does not change sign. However, depending on the amount of vanance in XY position of the surface, the change in direction may result in a negligible vaπation in cross-sectional position As can be seen, the slope of the part surface, identified as S' in the figure, changes sign over, for example, 10 layers. For a given number of layers, the thinner the layers the more likely the assumption will hold. If these above assumptions are made, the following formulas can be used to reduce the mathematical calculataons required: (7) ' C - = (P n -( + P π - - P n )' P i l n (8) B = P - P - C - P (9) S = T - P - C - B Instead of being based on the boolean summation of the area of every cross-section within a region, as oπgiπal equations (1) to (4), these equataons utilize the cross-sectional information from only the top and bottom cross-sections of the region If the assumptions always hold true, these formulas yield exact results. In any case, in practice, they have been shown to be very good approximations. It should be appreciated that to perform the aforementioned calculations, it is necessary to have available simultaneously the data from (t+u+1) layers (for example for t=10, u=5, we need data for 16 layers). That is because the support data for layer "n" is dependent upon the part and total data for layers "n+1" through "n+t", layers "n-1" through "n-u ", and of course for layer "n" To maintain such data in immediately accessible form, it is advantageous to use a πng buffer. As shown in Figure 36, a ring buffer is a circular buffer in which is stored the part and total data for t+u+1 layers (e.g. 16 layers). Figure 36a illustrates the state of the buffer in terms of a 16 layer (t=10, u=5) example when the calculations for layer n are about to be performed. A pointer, identified as PTR in the figure, is used to point to the current layer under consideration. As indicated, the data for layers "n+1" through "n+10", "n", and "n-1" through "n-5 " is stored in the buffer. A second pointer, identified as LAST in the figure, is used to point to the last entry in the buffer, in this case, the entry is for layer n-5. After the computations for layer "n" have been completed, it is necessary to update the buffer in preparation for performing the computetaons for layer "n+1" To accomplish this, PTR is first updated so that it points to the data for layer "n+1" Then, the data pointed to by LAST is overwritten by the data for the next layer to be added to the buffer, in this case layer "n+11" Finally, LAST is updated to point to the data which is now the last entry in the buffer, in this case, the data is for layer "n-4" The result of these three computations is illustrated in Figure 36b Figure 36c illustrates the status of the buffer at the point when the computetaons for layer n+2 are about to be performed This process then repeats until the computations for all the layers have been completed A number of alternative embodiments are possible for manipulating 3D object data into data useful for driving an SDM apparatus For example, in one alternate embodiment, the aforementioned calculations are performed using Boolean operations on polygonal data instead of on transitional data In another, the data for all the layers of the part is stored simultaneously in a memory instead of in a ring buffer In still another it is possible to equalize the rates of accumulataon of the thin, fiber-like supports and the part by employing multiple passes of the pπnt head It should also be appreciated that it is possible to compute bridge data or transitional support data from up-facing surfaces i e B„(U) This data could be used to form transitional supports between the thin fiber like column supports starting at an up-facing surface of the object and the 3 x 3 column supports sitting thereon Moreover, it should also be appreciated that it is not necessary to compute C π (U) data separately from C n (D) data if the Style file for the two is the same Of course, if the two Style files are intended to be different, then both categories of data should be maintained It should also be appreciated that it is possible to build an arbitrary number of support types or categories on a given layer using the subject invention instead of the three that have been discussed This can be accomplished simply by adding additional Style files and equataons for determining the areas in which the new categories of supports are to be built Build Styles and Support Styles: For optimum date handling it is advantageous not to embed regular pattern formations into the RLE data since this would make the RLE files excessively large and make data handling in a timely manner impractical As such it is advantageous to maintain object and support cross-sectaon information independent of exact exposure patterns (i e deposition pattems) until layer pπnting is to occur As mentioned above, at an appropriate time, the cross-sectional data (e g in the form of RLE information) is boolean intersected with the appropriate build style pattems to define the exact pattern that will be used to define the deposition detail For example, this can be used to create checkeΦoard pattems on a rapid basis An example of this is illustrated in Figures 40a-40c, in which like elements are referenced with like identifying numerals Figure 40a illustrates the desired image 28 to pπnt As shown, the desired image consists of two components The first component, identified with numeral 29, is a solid The second component, identified with numeral 30, is desired to be formed with an on-off checkeΦoard pattem For the reasons discussed, it may be prohibitively slow and memory intensive to convert the image 30 to a honeycomb pattem on a pixel by pixel basis Further manipulations of the data for image 30 may be unduly complicated and slowed by putting it in a honeycomb patte too early Transfer of date to a storage device (i e a hard disk or tape drive) may also be unduly encumbered by holding it in such a detailed format Thus, as shown in Figure 40b, the data for both pattems is maintained or converted into solid form (minimum transitions) for further manipulation whereafter it is transmitted to a digital signal processor which is responsible for controlling jetting and X,Y,Z motion Then as shown in Figure 40c, the data 31 associated with component 30, which is in solid form, is logically "ANDed" (i e boolean intersected) with honeycomb/checkerboard pattem 32 in order to change the solid data into the desired modulated form representative of the modulated cross-sectional pattern to be jetted Once in this final modulated form it is preferred that no further storage of the data occur but instead be used to control the firing of jets with or without further manipulation In this example the data for component 29 and 30 must now be "OR ed" together to yield the a single bit map containing the entire desired set of data It is this combined data which is then used to drive the finng of the printhead Data provided with the RLE file to the modeler includes building/support pattern style information for use as discussed above As discussed above the association of RLE data with modulation data is accomplished through the use of Style files each of which stores a particular "style" or building pattern Examples of building patterns are shown in Figures 41a, 41b, and 41c Figure 41a illustrates a checkerboard building pattern appropriate for use in building a category of supports as described in U S Patent Application 08/534,813 Figure 41b illustrates a pattem appropriate for use in building a second category of supports as also descnbed in U.S Patent Application 08/534,813 Figure 41c illustrates a pattern which specifies that solid be build Many other build styles are possible including multiple exposure build styles Such as the examples depicted in Figure 41d, in which altemate spaced scan lines are solidified in successive passes In this example pattem 56 is exposed dunng a first pass and pattem 57 is exposed in a second pass Another example is shown in Figure 41 e, in which altemate spaced columns are solidified on successive passes In this example pattern 58 is exposed during a first pass and pattem 59 is exposed during a second pass A third example is illustrated in Figure 41 f, in which non- overlapping checkerboard patterns are solidified on successive passes Pattem 60 is exposed in a first pass and pattern 61 is exposed in a second pass To associate different style files with different object and support regions, the RLE format is made to include a build pattern designation for each different sets of raster line transition information passed to the modeler The conceptual format of the RLE file is depicted in Figure 47 Through this file format, a user can specify virtually any building pattern for a given pair or pairs of transitional points Data Skewing In addition to providing a bit map containing the correct pixel information for controlling the firing of jets, the data must be readily extractable from the bit map and provided to the firing mechanism in the right order This need to place the data in an extractable form bπngs us to the next step in the data manipulation process This next step is called skewing For example the data can be processed so that the necessary information is available to allow the jets to simultaneously fire even though adjacent jets may not be located on adjacent raster lines or even simultaneously located, on their respective y raster lines, above the same X-coordinate. As such skewing refers to a data realignment process which is required, for example when the scan head is placed at an angle to the scanning direction (as depicted in Figure 2b), when multiple heads are used and are to be fired simultaneously or in sequence, or simply due to the jets not being spaced over aαjacent raster lines. In Figure 2b. for example, orifices 10(3) and 10(4), which are aligned in Figures 2a. become displaced in the scan direction by a distance d", as shown in Figure 2b, when the scan head is angled relative to the scan direction However the data used in relation to the configuration of Figure 2a would require that jets 10(3) and 10(4) fire at the same time to hit similar X locations With the configuration of Figure 2b. a distortion would be caused by use of such data Consequently, the data must be skewed, in this example, to correct for this relative displacement Tne problem is that the amount of data involved is relatively large, and the skewing must be performed in real time For example, an ink jet tn a typical configuration might take only 500 nS to pass over a given pixel Thus, any skewing process that operates on individual pixels cannot take longer than this time per pixel (on average) in order to keep up with the data consumption rate A typical digital signal processor,, e g a C31 processor, running at 40MHz. has a cycle time on the order of 50nS Therefore, if the time over any pixel location is on the order of 500 nS.there are only 10 cycles available to operate on a given pixel. Each processor instruction, on the other hand, requires a minimum of 1 cycle Often, several cycles are required to contend with bus conflicts, pipeline conflicts, and memory wait states Thus, each instruction may effectively require 2-4 cycles. Thus, only about 3 instructions can realistically be devoted to each pixel. The problem is that to perform a typical operation, such as setting an individual pixel to a logical "1", requires about 6 instructions Thus, it is not feasible to perform operations on a pixel by pixel basis. Instead, operations that operate on multiple pixels at a time, such as 32 pixels, are required Some typical operations might include cleanπg the image, moving the image, outputting the image, "AND"ing two images together, or "XOR"ιng two images together These types of instructions typically require fewer instructions (2 or 3 instead of 6) and operate on 32 pixels at one time Overall, they operate about 100X faster than operations on individual pixels As discussed above, a control computer performs the functions of slicing an STL or CTL file, and computing .RLE data for the various cross-sectaons A Digital Signal Processor (DSP) coupled to the print head must take this RLE data, decompress it, skew the data according to the jet arrangement, and then output the data to the jets As discussed, "skewing" refers to the process of manipulating the image data to compensate for the jet arrangement and possibly other factors Since the data, once decompressed, may not be able to be manipulated rapidly enough, it is advantageous to be able to manipulate the data while it is still in compressed form (e.g. while it is still in the .RLE format). Another cntical time-saving preference is that the data be stored in memory such that a 2- byte or 4-byte word contains pixels that are each desired to be output at the same time The process of skewing the data then involves simply shifting the start and stop transitions in the scan direction by an appropriate amount while keeping the data associated with pixels to be output at the same time in the same word The data is then decompressed, and individual words are sent to the pnnt head when the appropπate location in the X-direction is encountered The technique is illustrated in Figures 42a, 42b 42c, 42d, and 42e, in which like elements are referenced with like identifying numerals Figure 42a illustrates the pixelized image of the oπgiπal cross-sectaon Figure 42b illustrates this data in RLE format As shown, the date for the individual scan lines, identified in the figure with numerals 25(1), 25(2), 25(3), , 25(10), has been compressed into data representative of start and stop transitions Figure 42c illustrates the process of skewing this data to adjust for a pπnt head which is angled relative to the scan direction In this figure, it is assumed that the print head has 5 jets, and is angled such that the individual jets are relatively displaced from successive jets by one pixel Thus, the data for scan line 25(2) is displaced 1 pixel in relation to scan line 25(1), the data for scan line 25(3) is displaced 1 pixel in relation to scan line 25(2), etc The process continues until scan line 25(6) is encountered Since that is the sixth scan line, and will not be scanned on the same pass as the first 5 lines that line is not displaced relatave to the others Instead, scan line 25(7) is displaced 1 pixel relative to scan line 25(6) Scan line 25(8) is displaced 1 pixel relative to scan line 25(7) Scan line 25(9) is displaced 1 pixel relative to scan line 25(8), etc During this process, the skewed date is "banded" such that date associated with firings which are to occur at the same time is collected into a single word This data is then successively decompressed one band at a time The process is illustrated in Figure 42d The data for the pixels in each of the columns 27(1 ), 27(2), 27(3), , 27(12), each represent data which is to be fired at the same time Accordingly, each of these columns of date is stored in individually accessible words, and is thus simultaneously accessible A banding index 26 is also maintained to step through the data one column at a time As each column is encountered, it is decompressed in turn (i e each transition is converted to an on/off bit, e g 32 bits at a time) With reference to Figure 42d. for example, the banding index is located at column 27(8) Accordingly, as shown, the data in that column is decompressed The remaining data in columns 27(9) to 27(12) is still in compressed format However, as discussed, that date will be decompressed as it is encountered by the banding index Next, the date is sequentially output to the pnnt head, one column at a time The process is illustrated in Figure 42e As shown, the banding index has been reset and then used to successively step through the columns 27(1)-27(12) a second time As shown, the index is currently situated at column 27(5) Accordingly, the data in that column is output to the print head The data in the remaining columns 27(6)-27(12) will be output in turn Time of flight and Jet Firing: Before the above generated date results in the deposition of droplets of mateπal at the desired locations one critical function remains to be performed As the data is loaded into the ink jet head for finng, the system must determine when the ink jet head has reached the proper locataon to drop its material The proper finng time, as discussed in previously referenced U S Patent Application 08/534,813 , actually occurs sometime before the head is positioned over the proper deposition location This early-firing compensation is called the time of flight correction However, the system still must determine when it is at the appropπate location to issue the early firing signal The details of this determination process are given below To enable building with desired scan line resolution it is important to be able to fire the jets at any desired position along the scan direction This can be problematic when using an encoder to indicate actual X position wherein the encoder may not have fence triggers at the required positions In fact the encoder may be of lower resolution than that which is desired for printing As higher resolution encoders are more expansive and it is desired to keep equipment costs down and as it is a disadvantage to be limited to a single resolution or resolutions which are multiples of the fence spacing Other means are desirable for determining accurate finng positions Accurate firing positions as explained below are determined by performing a distance interpolation between fence lines based on a calculated average velocity and a known elapsed time since the last fence was passed The firing locations are then determined using the known desired firing point and the interpolated estimate of actual position The X-stage 12 (see Figure 1 ) has associated with it an encoder for use in determining the position of the print head in the X-direction so that the firing pulses for the print head can be initiated at the appropriate time In a preferred embodiment, to perform this function, a glass plate, identified with numeral 34 in Figure 43, is employed on which lines 33 are etched which are spaced from one another by 10 microns A light and photodiode detector (not shown) are also employed to determine when these lines are passed and to interrupt the DSP every time the print head passes one of these lines. A pair of detectors (not shown) is also employed to indicate whether the pπnt head is moving left or right To avoid bothering the DSP with signals caused by vibration and the like, a digital hysteπsis circuit (not shown) is employed to shield the DSP from spurious interrupts caused by vibrations and the like From this circuitry, it is possible for the DSP to determine the position of the print head within 10 microns, and also to determine the direction of movement In order to pπnt at a finer resolution than 10 microns, a counter is provided within the DSP to start counting whenever the DSP passes one of the aforementioned lines When the counter reaches a certain value, the DSP causes a firing signal to be generated to trigger the print head A second counter is also provided to deal with the situation illustrated in Figure 44 The signals T 0 , T-, T 2 , T 3 , and T„, identified with numeral 35, represent signals generated by the encoder from passage of the pnnt head past the lines 33 illustrated in Figure 43 The lines identified by numeral 36, in contrast, indicate the desired firing positions For the signals T 0 ', T , T 2 ', T 3 ', these signals all follow corresponding signals. T 0 , T,, T 2 , T 3 , respectively Thus, a single counter can be used in the generation of these signals in the manner described The problem that occurs is illustrated by signals T < and T 4 ' Since T Λ ' actually precedes its corresponding signal T 4 , a second counter must be provided for generating this signal in response to the occurrence of signal T 3 An algorithm for generating the firing signals is illustrated in Figures 45a-45b As shown, an interrupt, identified in Figure 12a with numeral 57, is generated as the print head passes by one of the encoder lines Then, in step 38, an encoder timer (not shown) is read and associated with the pπnt head position This step is performed over several encoder lines The resulting data is stored In step 39, the average velocity of the pπnt head is calculated from the stored data by dividing the change in position by the change in time over the prescπbed encoder lines In step 40, the distance, ΔD, between the next firing locataon and the last encoder line is determined In step 41, this value is used to calculate the time differential Δt(1 ), from the last encoder line until the next finng location taking into account left/right compensation and time of flight compensation Then, in step 42, this value is loaded into a first firing timer which as discussed, initiates a finng pulse when the same has expired In step 43 (Figure 45b) the time differential, Δt(2), for the next firing position is calculated in the manner described in relation to Δt(1 ) In step 44, this value is checked to see if the next finng position is located beyond the next encoder line If so, then that finng pulse can be initiated off of the next encoder line If not in step 45 that value is loaded into a second firing timer In step 46 a return from interrupt is then initiated Alternative embodiments may be used for linking encoder position to the issuing of finng commands One such alternative uses multiple encoder fence location time signals to derive a more accurate representation of the average velocity of the scanning head In this preferred embodiment, the last 8 encoder fence locations time signals are averaged to yield a time signal which can be associated with the position of the 4th encoder fence back The previous 8 encoder fence locations time signals are averaged to yield a time signal which can be associated with the 12th encoder fence back These two averaged time signals are used to derive an averaged velocity value for the scanning of the pπnt head From a determination of the distance between the 4th encoder back and the next firing locataon, the average velocity, the elasped time since the 4th encoder fence back was crossed, a time when the jet will reach the correct finng locataon is estimated, a timer is started using the estimated time, and the jet is fired when the time interval has elapsed This completes a discussion of the basic firing position enhancement algoπthm It should be appreciated that various enhancements or modifications are available, including compensation based on acceleration of the print head, or the use of more than one firing counter to further increase print resolution relatave to the increase achievable through two counters While embodiments and applications of this invention have been shown and described, it should be apparent to those skilled in the art that many more modifications are possible without departing from the inventive concepts herein The invention therefore, is not to be restricted, except in the spiπt of the appended claims

Claims: What is claimed is 1 A rapid prototyping apparatus compπsing a dispenser for controUably dispensing a flowable material which is solidifiable upon being dispensed a platform for supporting a cross-section of a three-dimensional object and providing a working surface for building a next object cross-section at least one dexer coupled to the dispenser and to the platform for relatively displacing the dispenser and the working surface in at least two dimensions comprising a scanning direction and an index direction and a controller coupled to the mdexer and to the dispenser for causing material to be dispensed over the working surface in accordance with a selected style 2 The apparatus of claim 1 wherein the apparatus is a selective deposition modeling apparatus and the dispenser is configured to selectively dispense material in accordance with the selected style 3 The apparatus of claim 2 wherein the controller is configured to provide said style which is a building style 4 The apparatus of claim 2 wherein the controller is configured to provide said style which is a support style 5 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies overprinting in the scan direction 6 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies higher resolution in the scan direction than the index direction 7 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies a higher drop density ratio for down-facing skin surfaces than for interior regions of the object 8 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies extending a down-facing skin region a plurality of layers above a down-facing surface 9 The apparatus of claim 8 wherein the controller is configured to provide said building style which specifies extending said down-facing skin region 5 layers above said down-facing surface 10 The apparatus of claim 8 wherein the controller is configured to provide said building style which specifies extending said down-facing skin region 10 layers above said down-facing surface 11 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies a higher drop density ratio for forming up-facing skin surfaces than for inteπor portions of the object 12 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies extending an up-facing skin region a plurality of layers below an up-facing surface. 13 The apparatus of claim 12 wherein the controller is configured to provide said building style which specifies extending said up-facing skin region 5 layers below said up-facing surface 14 The apparatus of claim 12 wherein the controller is configured to provide said building style which specifies extending said up-facing skin region 10 layers below said up-facing surface 15 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies using a higher drop density ratio for forming boundary regions of the object than for forming interior regions of the object 16 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies extending a boundary region a plurality of drop widths into an interior of the object. 17 The apparatus of claim 16 wherein the controller is configured to provide said building style which specifies extending said boundary region at least 2 drop widths into the interior of the object. 18 The apparatus of claim 16 wherein the controller is configured to provide said building style which specifies extending said boundary region at least 4 drop widths into the inteπor of the object 19 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies forming an interior region of the object with checkerboard supports 20 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies forming an inteπor region of the object with line supports. 21 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies forming an inteπor region of the object with column supports. 22. The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies forming an interior portion of the object with arch supports 23 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies changing the scan direction between two layers 24 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies reversing the scan direction between two layers 25 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies reversing the index direction between two layers 26 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies altering the scan and index directions between two layers 27 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies reversing the scan and index directions between two layers 28 The apparatus of claim 3 wherein the dispenser comprises at least one multi-jet ink-jet dispensing head 29 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies forming the object through raster scanning 30 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies forming the object through raster scanning having a length and width limited to a dispensing region required by a layer being formed 31 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies forming the object through vector scanning 32. The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies forming an interior region of the object through raster scanning, and a boundary region of the object through vector scanning 33 The apparatus of claim 28 wherein the controller is configured to provide said building style which specifies randomizing from layer to layer the jets which dispense on any XY locataon 34 The apparatus of claim 28 wherein the controller is configured to provide said building style which specifies printing using all jets a test pattern, and detecting therefrom jets which are not firing properly 35 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies maintaining a temperature of the object above a minimum temperature to reduce curl distortion 36 The apparatus of claim 3 wherein the controller is configured to provide said building style which specifies separately forming different components of the object to enable object surfaces to be reoriented as up-facing surfaces duπng part building, and then combining the separately formed components 37 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies checkeΦoard supports 38 The apparatus of claim 4 wherein the controller is configured to provides said support style which specifies employing a higher number of passes per layer to form supports than to form the object 39 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies drop-on/drop-off checkerboard supports 40 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies line supports 41 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies straight line supports 42 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies curved line supports 43 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies broken line supports 44 The apparatus of claim 4 wherein the controller is configured to provide said support style specifies supports with shelving on at least part of a layer 45 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies offsetting supports on layers above shelving 46 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies shelving on less than 10 consecutive layers 47 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies shelving on less than 5 consecutive layers. 48 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies shelving on a whole first layer. 49 The apparatus of claim 4 wherein the controller is configured to proviαe said support style which specifies shelving in X-Y regions not shelved in a previous layer 50 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies complementary shelving on subsequent layers 51 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies column supports 52. The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies column supports with shelving 53. The apparatus of claim 4 wherein the controller is configured o provide said support style which specifies offset column supports 54. The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies N-by-N column supports defined in terms of drop width 55. The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies N-by-N column supports defined in terms of pixels 56. The apparatus of claim 55 wherein N is 2. 57. The apparatus of claim 55 wherein N is 3. 58. The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies changing the support pattem in regions at least N layers below a down-facing surface. 59. The apparatus of claim 58 wherein N is 4. 60. The apparatus of claim 58 wherein N is 9 61 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies decreasing the drop density ratio in regions approaching a down-facing surface. 62. The apparatus of claim 61 wherein the support style further specifies utalizing at least one layer of shelving when transitioning to lower drop density ratao supports. 63 The apparatus of claim 61 wherein the support style further specifies switching from column supports to checkerboard supports in regions when approaching a down-facing surface. 64 The apparatus of claim 4 wherein the support style further specifies changing the support pattern in regions greater than a predetermined number of layers above an up-facing surface 65 The apparatus of claim 64 wherein the predetermined number is 4 66 The apparatus of claim 64 wherein the predetermined number is 9 67 The apparatus of claim 64 wherein the support style further specifies lowering the drop density ratio in regions after leaving an up-facing surface. 68. The apparatus of claim 67 wherein the support style further specifies utilizing at least one layer of shelving in transitioning from higher to lower drop density ratio supports. 69 The apparatus of claim 68 wherein the support style further specifies switching from column supports to checkerboard supports after leaving an up-facing surface. 70 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies arch supports. 71. The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies air pressure supports 72. The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies diffeπng the support pattern at different areas of a cross-sectaon. 73 The apparatus of claim wherein the controller is configured to provide said support style which specifies displacing the support structure from a boundary of the object by a first predetermined number of pixels in the scan direction, and a second predetermined number of pixels in the index direction. 74 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies building supports with a material different than that used to from surfaces and boundary regions of the object. 75 The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies bulk dispensing supports on each layer after material used to form the object is selectively dispensed. 76. The apparatus of claim 4 wherein the controller is configured to provide said support style which specifies the use of a water soluble material to build supports. 77 The apparatus of claim 3 wherein the controller is configured to provide said build style which specifies uniform temperature building 78 The apparatus of claim 3 wherein the controller is configured to provide said build style which specifies subpixel firing. 79. The apparatus of claim 3 wherein the controller is configured to provide said build style which specifies sub-raster line finng. 80. The apparatus of claim 3 wherein the controller is configured to Drovide said build style which specifies the use of material with a black body radiator added. 81. The apparatus of claim 78 wherein the build style further specifies subpixeling through shifted time of flight data. 82. The apparatus of claim 4 wherein the support style specifies bigger droplets for supports than for the object. 83. The apparatus of claim 3 wherein the build style further specifies a mateπal having inserted interlaced thermal conductors. 84. The apparatus of claim 3 wherein the build style further specifies object sensitive interlacing. 85. The apparatus of claim 3 wherein the build style further specifies drop width compensation. 86. The apparatus of claim 3 wherein the build style further specifies over-printing width compensation. 87 The apparatus of claim 3 wherein the controller is configured to provide said build style which specifies a spacing between the orifice of the dispenser and the working surface which is large enough such that the droplets form semi-spheπcal drops upon impact 88 The apparatus of claim 4 wherein the support style specifies lagging support building behind part building by at least one layer to avoid distortion of the supports caused by planaπzation 89 The apparatus of claim 3 wherein the build style specifies planaπzatton through melting alone 90 The apparatus of claim 3 wherein the build style specifies planarization through melting in combination with scraping 91 The apparatus of claim 3 wherein the build style specifies planarization through melting in combination with scraping and rotation 92 A method for rapid prototyping comprising controUably dispensing a flowable material which is solidifiable upon being dispensed, supporting a cross-section of a three-dimensional object and providing a working surface for building a next object cross-sectaon, relatively displacing the dispenser and the working surface in at least two dimensions, comprising a scanning direction and an index direction, and dispensing material over the working surface in accordance with a selected style 93 The method of claim 92 further comprising selectively dispensing material over the working surface in accordance with the selected style 94 The method of claim 93 wherein said style which is a building style 95 The method of claim 93 wherein said style which is a support style 96 The apparatus of claim 1 additionally compπsing means for forming support structures which branch out beyond mateπal dispensed on an immediately preceding lamina wherein the branching results in more support structures contacting a down-facing object surface than the number of support structures from which the branching was initiated 97 The method of claim 92 additionally compπsing the step of forming support structures which branch out beyond material dispensed on an immediately preceding lamina wherein the branching results in more support structures contacting a down-facing object surface than the number of support structures from which the branching was initiated 98 The apparatus of claim 1 additionally comprising means directing droplets at a focal plane below a level of an actual working surface for achieving a self correcting accumulataon in a direction perpendicular to a plane of the cross-sections 99 The method of claim 92 additionally compnsmg the step of directing droplets at a focal plane below a level of an actual working surface for achieving a self correcting accumulataon in a direction perpendicular to a plane of the cross-sections 100 The apparatus of claim 1 additionally comprising means for directing a cooling gas onto the surface of a partially formed object and means for removing the cooling gas from the area above said surface 101 The method of claim 92 additionally comprising the step of directing a cooling gas onto the surface of a partially formed object and the step of removing the cooling gas from the area above said surface What is claimed is 102 In a rapid prototyping system, a method of generating start/stop transition date representative of object cross-sections, comprising the steps of accepting as input a three-dimensional data representation of object, associating slicing layers with object cross-sectaons, for each slicing layer intersecting three-dimensional data representation with a plurality of pixel scan lines, and generating the transition data for a cross-section at points of intersection between the scan lines and the three-dimensional object representataon 103 The method of claim 1 wherein the rapid prototyping system is a selective deposition modeling system 104 In a rapid prototyping system a method of generating data descriptive of a support region for a given layer of the object, comprising the steps of determining the total region covered by the combination of all layers above the given layer, taking the boolean difference between a region of the given layer with the total region; and using the difference as the support region for the given layer 105 The method of claim 3 wherein the rapid prototyping system is a selective deposition modeling system 106 In a rapid prototyping and manufacturing system a method for increasing resolution of a print head in a scan direction comprising the steps of providing a processor, providing first and second timers which, upon being loaded, will count down and initiate a firing pulse upon expiration, interrupting the processor when the pπnt head has passed by one of a prescribed sequence of encoder lines, calculating the average velocity of the print head over several encoder lines, using said average velocity to calculate a first value representative of the change in time from the one encoder line to a next firing location loading said value into said first timer, using said average velocity to calculate a second value representative of the change in time from the one encoder line to a second firing locataon subsequent to the next finng locataon, and loading said second value into said second timer if said second firing location is before the next encoder line. 107. The method of claim 5 wherein the rapid prototyping system is a selective deposition modeling system. 108. In a rapid prototyping and manufacturing system, a method of assigning build patterns to different portions of a cross-section to be solidified comprising the steps of: representing portions of the cross-sectaon to be solidified as pairs of start and stop transition data; assigning to a pairs of said data building patterns which modulates the actual starting and stopping locations to obtain modulated data; and dispensing material in accordance with said modulated data in order to build said cross- section 109. The method of claim 5 wherein the rapid prototyping system is a selective deposition modeling system. 110. In a rapid prototyping and manufacturing system, a method of increasing resolution in an index direction by skewing said data.

Inventors: ALMQUIST THOMAS A, BEDAL BRYAN J L, EARL JOCELYN M, FEDCHENKO RICHARD P, HULL CHARLES W, KEREKES THOMAS A, LEYDEN RICHARD N, LOCKARD MICHAEL S, MEROT CHRISTIAN M, PANG THOMAS H, SMALLEY DENNIS R, THAT DINH TON, THAYER JEFFREY S, ALMQUIST, THOMAS, A., BEDAL, BRYAN, J., L., EARL, JOCELYN, M., FEDCHENKO, RICHARD, P., HULL, CHARLES, W., KEREKES, THOMAS, A., LEYDEN, RICHARD, N., LOCKARD, MICHAEL, S., MEROT, CHRISTIAN, M., PANG, THOMAS, H., SMALLEY, DENNIS, R., THAT, DINH, TON, THAYER, JEFFREY, S.
Applicants: 3D SYSTEMS INC, 3D SYSTEMS, INC.
Application year: 1996
Main classification: B29C67
IPC codes: B22F3, B29C41, B29C67, B29K63, G06T17