Abstract:
Planar texture maps which reflect the distances and angles of a 3-D surface are generated. A user is permitted to manually adjust the balance between discontinuity and distortion. The user selectively modifies the 3-D surface, and by doing so adjusts the balance between discontinuity and distortion in the planar map. Each point on the 3-D surface corresponds to a unique point on the planar map. Operations may therefore be performed on the simpler 2-D planar map rather than the more complex 3-D map, and the result of the operations may be uniquely mapped to the 3-D surface. Further, the majority of the vertices on a 3-D surface are mapped automatically, even though the user maintains a high degree of control over the mapping process via altering the 3-D surface boundary. User-selected map vertices may be pinned to a user-selected location, and held fixed while a conventional relaxation technique is applied. This provides the user with a greater degree of control over the relaxation process.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates generally to computer graphics, and more particularly to the generation of planar maps of three-dimensional surfaces. 
     2. Related Art 
     The problem of flattening a three dimensional (3-D) surface into a two-dimensional (2-D) domain is age old and takes several forms. 
     For example, one of the central concerns of cartography is the representation of a sphere as a planar map. As is well known, it is impossible to represent such a surface in 2-D without distortion and discontinuity. In a Mercator projection, for example, Greenland appears much larger than it really is in relation to more southern countries. Cartographers mitigate this problem either by cutting the surface of the globe into segments, thus trading off increased discontinuity for decreased distortion, or by using other projections which trade off size distortions for shape distortions. 
     Other application areas involve surface flattening, or the inverse problem, the construction of 3-D surfaces from originally flat components. An example of flattening involves taking the hide of an animal and creating flat pieces of leather or fur. An example of the inverse process is the construction of apparel such as shoes and garments out of pieces of leather, fur, or cloth. Going in either direction involves stretching/shrinking (distortion) and the alteration of discontinuity/continuity (e.g., by cutting or sewing). 
     The problem of correspondences between flat and curved surfaces arises in computer graphics software. A technique known as texture mapping is used to give 3-D surfaces character and realism. Whenever there exists a mapping from a 3-D surface to a 2-D region, an arbitrary image can be identified with the 2-D region so that rendered attributes of the surface (e.g., color, shininess, displacement) are controlled by the image&#39;s values. An image used in this way is called a texture map. 
     For 3-D surfaces such as bicubic patches, the mapping from surface to a 2-D region is intrinsic. However, for 3-D surfaces composed of polygons, an a priori mapping does not exist. The construction of such mappings, known as parameterization, has been a problem in computer graphics for several years because without a parameterization a polygonal surface is not amenable to texture mapping. Parameterization is equivalent to flattening. 
     Some conventional parameterization methods involve 1) selecting a boundary on a 3-D surface, 2) mapping the boundary to a planar convex polygon, and 3) using relaxation methods to calculate a mapping of interior points of the surface to interior points of the convex polygon. To visualize this process, imagine the 3-D surface to be a rubber sheet whose boundary is stretched around the perimeter of the polygon. 
     A drawback with these methods is that the planar convex polygon (e.g., a circle or square) does not necessarily reflect the shape of the surface boundary, thus increasing the possibility that the mapping will introduce shape distortions, particularly in polygons close to the boundary. A need therefore exists for an improved system and method for creating a planar boundary which inherits the geometry of a given surface boundary. 
     SUMMARY OF THE INVENTION 
     Briefly stated, the present invention is directed to a system and method for generating planar maps which reflect the distances and angles of a 3-D surface, where the user can manually adjust the balance between discontinuity and distortion. 
     A preferred embodiment of the present invention includes receiving a 3-D surface, defining a surface boundary on the 3-D surface, and generating a planar map based on the 3-D surface and the defined surface boundary. An edge-and-angle proportional mapping is preferably used to map the surface boundary to a map boundary. Those surface vertices not forming the surface boundary are then relaxed to create the map vertices not forming the map boundary. 
     A feature of the present invention is that a user selectively adjusts the balance between discontinuity and distortion in the planar map. 
     Another feature is that the present invention allows for the creation of planar maps such that each point on the 3-D surface corresponds to a unique point on the planar map. As a result, operations may be performed on the simpler 2-D planar map rather than the more complex 3-D map, and the result of the operations may be uniquely mapped to the 3-D surface. 
     Another feature of the present invention is that the majority of the vertices on a 3-D surface are mapped automatically, even though the user maintains a high degree of control over the mapping process via altering the 3-D surface boundary. 
     Another feature of the present invention is that user-selected map vertices may be pinned to a user-selected location, and held fixed while a conventional relaxation technique is applied. This provides the user with a greater degree of control over the relaxation process. 
     Another feature of the present invention is that a map boundary interpolation can be performed to generate alternate planar maps. These alternate planar maps may be superior in some respect to non-interpolated planar maps. 
     Further features and advantages of the invention, as well as the structure and operation of various embodiments of the invention, are described in detail below with reference to the accompanying drawings. In the drawings, like reference numbers generally indicate identical, functionally similar, and/or structurally similar elements. The drawing in which an element first appears is indicated by the leftmost digit(s) in the corresponding reference number. 
    
    
     BRIEF DESCRIPTION OF THE FIGURES 
     The present invention will be described with reference to the accompanying drawings, wherein: 
     FIG. 1 is a block diagram of a digital computer environment within which the present invention is used; 
     FIG. 2 is a flowchart illustrating the functions performed according to the present invention; 
     FIG. 3A depicts an example 3-D surface; 
     FIG. 3B depicts an example 3-D surface with a defined boundary; 
     FIG. 3C depicts an example 3-D surface with a modified (cut) boundary; 
     FIG. 4A depicts an example planar map of the example 3-D surface with no cuts; 
     FIG. 4B depicts an example distortionless planar map of the example 3-D surface after cuts; 
     FIG. 5 is a flowchart detailing the generation of planar maps; 
     FIG. 6 is a flowchart detailing boundary mapping; 
     FIG. 7 is a flowchart detailing a preferred embodiment of an edge-and-angle proportional boundary mapping; 
     FIG. 8A illustrates a first step in a preferred embodiment of an edge-and-angle proportional boundary mapping; 
     FIG. 8B illustrates a second step in a preferred embodiment of an edge-and-angle proportional boundary mapping; 
     FIG. 8C illustrates a third step in a preferred embodiment of an edge-and-angle proportional boundary mapping; 
     FIG. 8D depicts a map boundary generated according to a preferred embodiment of an edge-and-angle proportional boundary mapping; 
     FIG. 9 is a flowchart detailing relaxation and pinning; 
     FIG. 10A depicts an example planar map with a distorted map primitive; 
     FIG. 10B depicts an example planar map after a map vertex has been moved and pinned; and 
     FIG. 11A depicts an example self-crossing map boundary generated according to a preferred embodiment of an edge-and-angle proportional boundary mapping (source boundary); 
     FIG. 11B depicts an example map boundary generated according to an edge-proportional boundary mapping (target boundary); 
     FIG. 11C depicts an example map boundary generated according to an interpolation between a source boundary and a target boundary; and 
     FIG. 12 is a flowchart detailing the map boundary interpolation method. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Overview of the Environment 
     FIG. 1 is a block diagram of a digital computer environment  100  within which the present invention is used. A user  102  interacts with a computer system  104  according to conventional user interface techniques. The present invention is preferably implemented as computer readable program code  108  stored on a computer usable medium  110 , accessed via a communication link  112 . Computer system  104  operates under the control of computer readable program code  108 . 
     Computer system  104  represents any computer system known to those skilled in the art with sufficient capability (i.e., both memory and processing power) to execute computer readable program code  108 . Examples of computer system  104  include, but are not limited to, a microcomputer, workstation, or terminal connected to a central processor. 
     Computer usable medium  110  is any digital memory capable of storing computer readable program code  108 . Examples of computer usable medium  110  include, but are not limited to, hard disks (both fixed and portable), floppy disks, and CD-ROMs. 
     Communication link  112  provides the communication pathway between computer system  104  and computer usable medium  110 . Communication link  112  includes, but is not limited to, parallel cables, local area networks, wide area networks, the Internet, and any other combination of electrical equipment designed to pass an electrical signal from one point to another. 
     User  102  communicates with computer system  104  according to conventional user input/output (I/O) techniques. Computer system  104  (under the control of computer readable program code  108 ) provides output to user  102  according to the present invention via video displays and audio messages, for example. User  102  provides input to computer system  104  via a keyboard, mouse, joystick, voice recognition, or any other type of input device known to those skilled in the art. 
     The present invention is preferably implemented as computer readable program code  108 . Computer readable program code  108  provides the instructions necessary for computer system  104  to execute the functionality described below. Those skilled in the art will recognize that computer readable program code  108  might be implemented in any computer language (e.g., C) acceptable to computer system  104 . 
     The present invention is described below in terms of computer graphics displays and operations performed upon these displays. Example graphical displays are provided, along with flowcharts which describe various operations performed on the displays. Those skilled in the art will readily recognize how to implement the graphical displays and operations on those displays as computer readable program code  108 . 
     Over View of the Invention 
     The purpose of this section is to provide an overview of the functionality encompassed by the present invention. FIG. 2 is a flowchart  200  describing functions performed by computer system  104  under the control of computer readable program code  108  according to the present invention, and the relative order in which these functions preferably occur. Those skilled in the art will recognize that these steps might equivalently be performed in different sequence, and still achieve the results described below. 
     In step  202 , computer system  104  receives a 3-D surface. FIG. 3A depicts an example 3-D surface  300 . The functionality of the present invention is described below as operations on 3-D surface  300 —those skilled in the art will recognize how these operations would apply to any arbitrary 3-D surface. Those skilled in the art will also recognize that 3-D surface  300  can be received from any digital memory accessible via communication link  112 , not just computer usable medium  110 . 
     3-D surface  300  includes one or more surface primitives  302 . A surface primitive  302  is a planar region bounded by surface edges  304 . A surface vertex  306  is the point at which two or more surface edges  304  meet. Consider 3-D surface  300 . Surface vertices  306  are labeled with a single digit number between 1 and 8, and will be referred to, e.g., as surface vertex 1. Surface edges  304  will be referred to herein with reference to the surface vertices  306  each particular edge connects. For example, surface edge  304  connecting surface vertex 2 and 6 will be referred to as surface edge 2-6. Similarly, surface primitives  302  will be referred to with reference to the surface vertices  306  surrounding each particular surface primitive. For example, surface primitive  302  bounded by surface vertices 1, 2, 6, and 5 will be referred to as surface primitive 1-2-6-5 (shown in FIG. 3A as the shaded region). 3-D surface  300  is an “open-topped” box-surface vertices 1 and 3 are not connected, and surface primitives 1-2-3 and 1-3-4 do not exist. 3-D surface  300  therefore includes five surface primitives  302 , eight surface vertices  306 , and twelve surface edges  304 . 
     Returning to FIG. 2, in step  204 , a surface boundary is defined on 3-D surface  300 . The surface boundary is used by the present invention to generate a planar map from 3-D surface  300 , as is described in detail below. A surface boundary is a closed path along 3-D surface  300  composed of surface edges  304  and surface vertices  306 . In FIG. 3B, four surface edges (4-1, 1-2, 2-3, and 3-4) and four surface vertices (1, 2, 3, and 4) form surface boundary  308 , which will be referred to herein as surface boundary 1-2-3-4. 
     In step  206 , a planar map is generated based on the 3-D surface received in step  202 , given the surface boundary defined in step  204 . FIG. 4A depicts an example planar map  400 . As with 3-D surfaces, planar map  400  includes map primitives  402 , map edges  404 , map vertices  406 , and a map boundary  408  made up of four map edges (1-2, 2-3, 3-4, and 4-1) and four map vertices (1, 2, 3, and 4). However, planar map  400  is defined in two dimensions (i.e., planar map  400  is defined in a single plane) whereas 3-D surface  300  is defined in three dimensions. The same naming conventions described above with respect to 3-D surface  300  will be followed with respect to planar map  400 . 
     Planar map  400  must satisfy the following restrictions: each point on 3-D surface  300  must map to a unique point on planar map  400 , and the connectivity of 3-D surface must be maintained in planar map  400 . The uniqueness restriction ensures that each element (i.e., vertex, edge, and primitive) of the 3-D surface corresponds in a one-to-one fashion with one element of the planar map. Map vertices  406  correspond in one-to-one fashion with surface vertices  306 , map edges  404  correspond to surface edges  304 , and so on. For example, map vertex 2 corresponds to surface vertex 2, map edge 2-1 corresponds to surface edge 2-1, map primitive 1-2-6-5 corresponds to surface primitive 1-2-6-5 (both are represented as shaded regions), and map boundary 1-2-3-4 corresponds to surface boundary 1-2-3-4. In order to satisfy this restriction, none of the map primitives  402  may overlap, otherwise the point of overlap would correspond to multiple locations on 3-D surface  300 . However, it is not required that shapes or sizes be maintained, i.e., a map edge may be longer or shorter than the corresponding surface edge and a map primitive may be of a different shape and size than the corresponding surface primitive. 
     It is this one-to-one correspondence that makes planar map  400  a useful representation of 3-D surface  300 . Consider an embodiment where planar map  400  represents a region of a texture map for a 3-D graphical surface in a computer graphics system. When planar map  400  satisfies the above restriction, the texture region within map primitive  402  will appear only on the corresponding surface primitive  302 , though the texture may be distorted due to any differences in shape between the primitives. Those skilled in the art will recognize additional advantages of ensuring this unique one-to-one correspondence between the 3-D surface and the planar map. 
     In addition to requiring one-to-one correspondence, planar map  400  must also maintain the connectivity of 3-D surface  300 . That is, map edges  404  and map vertices  406  must be interconnected in the same way that the corresponding surface edges  304  and surface vertices  306  are interconnected. For example, in FIG. 3B, surface vertex 4 is connected (via a single surface edge) to surface vertices 1, 3, and 8. Map vertex 4 must therefore be connected (via a single map edge) to map vertices 1, 3, and 8, as shown in FIG.  4 A. Again, it is not required that the edges be the same length, nor is it required that the primitives be the same shape. 
     Returning to FIG. 2, flowchart  200  indicates that program flow proceeds from step  206  back to step  204 . According to the current invention, the surface boundary  308  defined in the first iteration of step  204  can be modified in subsequent iterations. Once surface boundary  308  has been modified, a new planar map is generated in step  206 . This iteration may be repeated as many times as necessary to achieve a desired planar mapping of the received 3-D surface. 
     The following sections provide detailed descriptions of steps  204  and  206 . Defining the surface boundary is described first, followed by the generation of planar maps. 
     Defining the Surface Boundary 
     In step  204 , a surface boundary is defined on the 3-D surface. The 3-D surface received in step  202  may have a pre-defined surface boundary, depending on the particular application. Thus, on a first iteration through the method illustrated in flowchart  200 , the surface boundary may need to be defined in its entirety (if the 3-D surface has no pre-defined boundary), or user  102  may wish to modify the pre-defined surface boundary if one is present. On subsequent iterations, in step  204  user  102  may modify the surface boundary defined in the previous iteration. Thus, step  204  encompasses specifying surface boundary  308  in its entirety and/or modifying an existing surface boundary  308 . 
     Where a surface boundary must be specified in its entirety, user  102  preferably traces the desired surface boundary  308  directly onto the displayed image of 3-D surface  300  using a mouse, trackball, joystick, or any other appropriate user input device. Those skilled in the art will recognize that standard graphical user interface (GUI) techniques might be employed to implement this tracing function. However, those skilled in the art will also recognize that methods other than manual definition could be used for defining surface boundary  308 , such as an automated method which defines surface boundary  308  based on some arbitrary criteria. 
     Once surface boundary  308  has been defined, user  102  thereafter preferably modifies surface boundary  308  using “cutting” and “sewing” operations. Cutting 3-D surface  300  refers to splitting a surface edge adjacent to surface boundary  308 , such that the modified surface boundary  308  now traces through the split edge. For example, FIG. 3C depicts 3-D surface  300  with a surface boundary  308  modified by a cutting operation. Here, user  102  has cut 3-D surface  300  from surface vertex 4 to surface vertex 8. Cutting operations will be described herein with reference to the surface vertices through which the cut passes. For example, the cut depicted in FIG. 3C would be described as a cut 4-8. 
     As a result of the cutting operation, an additional surface vertex and surface edge has been created. Surface vertex 4 has been split into surface vertex 4a and 4b, and two surface edges (4a-8 and 4b-8) now exist where there was only one (4-8). The modified boundary on 3-D surface  300  is now indicated by surface boundary 1-2-3-4a-8-4b. FIG. 3C depicts surface vertices 4a and 4b as being spaced apart for illustrative purposes only. Surface vertices generated as the result of a cutting operation actually occupy the same point in 3-D space, as do the split surface edges. 
     Sewing refers to an operation which is the opposite of cutting, i.e., user  102  may rejoin surface edges and surface vertices which have been split as the result of a cutting operation. Sewing operations will be also described herein with reference to the affected surface vertices. 
     Cutting a 3-D surface allows, in many instances, a less distorted planar map to be generated. Consider FIGS. 3B and 4A. Planar map  400  is a 2-D mapping of 3-D surface  300 . As can be seen, the shapes of map primitives  402  are distorted as compared to their corresponding surface primitives  302 . This distortion is due to the restrictions placed on planar map  400  (i.e., uniqueness and connectivity). Those skilled in the art will recognize the desirability of producing planar maps where each map primitive accurately reflects the shape of the corresponding surface primitive. For example, map primitive 1-2-6-5 (shown as the shaded map primitive in FIG. 4A) would ideally have the same shape as surface primitive 1-2-6-5 (shown as the shaded primitive in FIG.  3 B). However, as can be seen, these shapes are similar but not identical. The measure of this difference in shape will be referred to as distortion. 
     Given the restrictions on planar map  400 , it is generally not possible to produce a distortionless planar map of a 3-D surface. Consider 3-D surface  300 . Clearly, it is not possible to “flatten out” this open-topped box to a distortionless planar map while maintaining connectivity and one-to-one correspondence. However, it would be possible if one were to cut the box along certain of the edges. The box could then be unfolded and laid flat, while maintaining connectivity and one-to-one correspondence. 
     FIG. 4B depicts a distortionless planar map  400  generated according to the present invention where user  102  performed the following cuts to 3-D surface  300  in step  204 : 4-8, 8-7, 8-5, and 5-6. As shown in FIG. 4B, map boundary 1-4b-8c-5b-6-5a-8b-7-8a-4a-3-2 results from this series of cuts. Now, for example, map primitive 4a-3-7-8a and surface primitive 4-3-7-8 have an identical shape, as do the other four primitives. Those skilled in the art will recognize that further cuts will not further decrease the distortion, as planar map  400  is already distortionless. Connectivity is also maintained, as each cut creates an extra surface vertex and surface edge, which may then be separated and laid flat. Thus, cutting operations in some instances will decrease distortion in planar map  400 . 
     However, distortion is not the only measure of interest. Those skilled in the art will also recognize that the discontinuities introduced by the cuts are not desirable for some applications. In the present context, a discontinuity refers to the situation where the map primitives corresponding to adjoining surface primitives do not adjoin because of a cut. For example, surface primitive 4-3-7-8 adjoins (i.e., shares an edge with) surface primitive 1-4-5-8; however, their corresponding map primitives 4a-3-7-8a and 6-7-8b-5a do not adjoin because of cut 7-8. Planar map  400  has greater discontinuity in FIG. 4B than in FIG. 4A, but less distortion. 
     This illustrates the fundamental tradeoff between discontinuity and distortion. Cutting 3-D surface  300  increases the discontinuity of planar map  400  but decreases the distortion. Those skilled in the art will recognize that the particular application to which the present invention is applied will determine the appropriate balance between distortion and discontinuity. The iterative approach of the present invention allows user  102  to achieve a desired balance by varying the number of cuts made to 3-D surface  300 . 
     Generation of Planar Maps 
     Once 3-D surface  300  has been received, and surface boundary  308  defined, a planar map is then generated in step  206 . FIG. 5 depicts a flowchart  500  illustrating step  206  in greater detail. Planar map generation is fundamentally a two step process: boundary mapping and relaxation. In step  502 , surface boundary  308  (see FIG.  3 B), which is defined in three dimensions, is mapped according to the present invention to 2-D map boundary  408  (see FIG.  4 A). In step  506 , conventional relaxation techniques are then applied to the remainder of the surface vertices (those surface vertices  306  not forming surface boundary  308 ) to form map vertices  406 . These steps are described in detail in the following two sub-sections. 
     Steps  504  and  508  describe an extension to the basic planar mapping described in steps  502  and  506 , whereby the map boundary created in step  502  is interpolated with another map boundary created according to a conventional boundary mapping technique. This extension is discussed in detail below. 
     Boundary Mapping 
     FIG. 6 depicts a flowchart  600  illustrating step  502  (mapping surface boundary  308  to map boundary  408 ) in greater detail. In step  602 , the surface angle is calculated at each surface vertex  306  on surface boundary  308 . For example, referring to FIG. 3B, the surface angle is calculated at surface vertices 1, 2, 3, and 4. The surface angle is calculated at each surface vertex  306  by determining the angle between the two boundary edges which meet at the surface vertex, where the angle is calculated traveling along 3-D surface  300 . 
     For example, at surface vertex 4, the surface angle is equal to the angle separating surface edges 1-4 and 3-4 traveling along 3-D surface  300 . To calculate the angle traveling along 3-D surface  300 , the angle between surface edges 4-1 and 4-8 must be added to the angle between surface edges 4-8 and 3-4. Each of these angles is 90 degrees, so the surface angle at surface vertex 4 is 180 degrees. Similarly, at surface vertex 1, the surface angle is the total of the angle between surface edges 1-4 and 1-5 (90 degrees), and the angle between surface edges 1-5 and 1-2 (135 degrees, totaling 225 degrees). The surface angle at surface vertex 2 is 90 degrees, and 225 degrees at surface vertex 3. 
     In step  604 , the length of each surface edge forming surface boundary  308  is calculated. For example, in FIG. 3B, the length of surface edges 1-4, 1-2, 2-3, and 3-4 are calculated. 
     In step  606 , map boundary  408  is created, where the planar angle at each map vertex  406  along map boundary  408  is proportional to the surface angle at the corresponding surface vertex  306 . Similarly, the length of each map edge along map boundary  408  is proportional to the length of the corresponding surface edge along surface boundary  308 . Step  606  is therefore called an “edge-and-angle proportional” mapping. 
     The term “proportional” as used herein indicates approximate proportionality by a factor common to all the planar/surface angle relationships, k a , and a factor common to all the map/surface edge relationships, k e . For example, the planar angle at map vertex 4 is approximately equal to k a  times the surface angle at surface vertex 4, the planar angle at map vertex 1 is approximately equal to k a  times the surface angle at surface vertex 1, and so on for the remainder of the corresponding planar/surface angle relationships along map boundary  408 . Similarly, the length of map edge 3-4 is approximately equal to k e  times the length of surface edge 3-4, the length of map edge 4-1 is approximately equal to k e  times the length of surface edge 4-1, and so on for the remainder of the corresponding map/surface edge relationships along map boundary  408 . 
     Those skilled in the art will recognize that various approaches might be followed for calculating planar angles and map edge lengths which are proportional to surface angles and surface edge lengths. FIG. 7 depicts a flowchart  700  which illustrates a preferred edge-and-angle proportional mapping method. Reference will also be made to FIGS. 8A-8D which graphically illustrate these operations on surface boundary  308  as defined in FIG.  3 B. Note that these illustrations are meant for explanatory purposes only—in a preferred embodiment, the calculations take place within computer system  104  and only the final result (FIG. 8D) is displayed to user  102 . 
     In step  702 , the planar angles of map boundary  408  are set equal to the corresponding surface angles calculated in step  602 , and the map edge lengths of map boundary  408  are set equal to the corresponding edge lengths calculated in step  604 . FIG. 8A depicts map boundary  408  at this stage of the edge-and-angle proportional mapping. The planar angle at each map vertex  406  is the angle between the two map edges  404  which meet at the map vertex in the plane in which map boundary  408  is defined. For example, map vertex 3 has a planar angle  802  as shown in FIG.  8 A. Planar angle  802  is set equal to the surface angle calculated at surface vertex 3, i.e., 225 degrees. Planar angle  808  indicates the angle between map edges 1-2 and 2-3—the dashed line indicates the angular position of map edge 2-3 relative to map edge 1-2—which is set equal to the surface angle calculated at surface vertex 2, i.e., 90 degrees. 
     In step  704 , the planar angles of map boundary  408  are adjusted in prorata fashion so that their sum is π(n−2), where n is the number of map edges forming map boundary  408 . The appropriate angle proportionality factor, k a , for achieving this angle sum is computed according to the following equation:          k   a     =       π                   (     n   -   2     )         α   tot                              
     where α tot =sum of planar angles on map boundary The planar angle at each map vertex along map boundary  408  is multiplied by k a ; proportionality, as described above, is therefore maintained. FIG. 8B depicts map boundary  408  after planar angles  802 ,  804 ,  806 , and  808  have been adjusted. Note that at this step in the mapping, the map edge lengths along map boundary  408  are equal to the corresponding surface edge lengths, and are therefore proportional (i.e., k e =1). 
     In FIG. 8B, note that map boundary  408  remains open. As noted above, one constraint on boundaries, both surface and map, is that they be closed. Therefore, in step  706 , map vertices  406  are appropriately displaced so as to close map boundary  408 . According to the present invention, map vertices  406  are displaced in a computationally efficient manner while maintaining edge and angle proportionality. 
     In a preferred embodiment, each map vertex  406  is displaced in the same direction, by an amount determined by the position of the vertex on map boundary  408 . Referring to FIG. 8C, map boundary  408  is shown as being open at map vertex 2 (shown between map edges 3-2 and 1-2, though by definition map vertex 2 only exists at the intersection of these two map edges). Map boundary  408  is closed by displacing the end of map edge 3-2 (at map vertex 2) according to a displacement vector  810 . The magnitude and direction of displacement vector  810  are equal to the magnitude and direction of the displacement necessary to close map boundary  408 . As shown in FIG. 8D, map vertex 2 is now properly displayed at the juncture of map edges 3-2 and 1-2. 
     The remainder of the map vertices on map boundary  408  are also displaced. In a preferred embodiment, each displacement vector is calculated according to the following formula:              D   _     i     =       (       L   i       L   tot       )          D   _                                             where                          D   _     i             displacement                 vector                 at                   i      th                   map                 vertex                                        D   _             displacement                 vector                 at                 map                 boundary                 opening                           L   i           length                 of                                map                 boundary                 at                   i      th                   map                 vertex                           L   tot           total                 length                 of                 map                 boundary                                
     Referring to FIG. 8C, {overscore (D)} is illustrated as displacement vector  810  (the displacement necessary to close map boundary  408 ), {overscore (D)} 1  is illustrated as displacement vector  816 , {overscore (D)} 4  is illustrated as displacement vector  814 , and {overscore (D)} 3  is illustrated as displacement vector  812 . 
     The remaining map vertices (3, 4, and 1) are displaced according to displacement vectors given by the above formula. L tot  is the total length of map boundary  408  (i.e., sum of the length of map edges 1-2, 1-4, 4-3, and 3-2). L i  is the length along map boundary  408  where map vertex i (i.e., the map vertex for which a displacement vector is being calculated) is positioned. For example, at map vertex 3, L 3  is equal to the sum of the length of map edges 1-2, 1-4, and 4-3. This sum is divided by the total length of map boundary  408  and multiplied by {overscore (D)} to determine {overscore (D)} 3  (displacement vector  812 ). At map vertex 4, L 4  is equal to the sum of the length of map edges 1-2 and 1-4. This sum is divided by the total length of map boundary  408  and multiplied by {overscore (D)} to determine {overscore (D)} 4  (displacement vector  814 ). 
     FIG. 8D depicts map boundary  408  after the map vertices have been displaced according to displacement vectors  810 ,  812 ,  814 , and  816 , as shown in FIG.  8 C. As required, map boundary  408  is now closed. Those skilled in the art will recognize that many different approaches might be taken to close map boundary  408 , and still achieve the same effect of maintaining edge and angle proportionality while minimizing computational costs. This closing of map boundary  408  completes step  706  in FIG. 7, step  606  in FIG. 6, and step  502  in FIG.  5 . 
     Those skilled in the art will also recognize that alternative approaches exist for creating an edge-and-angle proportional boundary in step  606 . One alternative approach is to simulate a system of rods whose lengths are proportional to the surface edge lengths. Each adjacent pair of rods is connected with an angular spring whose rest angle is the surface angle, and which delivers a force proportional to the departure from that angle. The simulated system is allowed to relax while constrained to a plane. The resulting configuration of rods is the map boundary. 
     Relaxation &amp; Pinning 
     Returning to FIG. 5, now that surface boundary  308  has been mapped to map boundary  408 , in step  506  a conventional relaxation technique is applied to the remainder of the surface vertices  306 , i.e., those surface vertices  306  not on surface boundary  308 . Relaxation techniques are well known to those skilled in the art, such as the techniques described in Matthias Eck et al., “Multiresolution Analysis of Arbitrary Meshes,” Computer Graphics (SIGGRAPH &#39;95 Proceedings), 1995, pp. 175-76 (ACM-0-89791-701-4/95/008), which is incorporated herein by reference. Relaxation may be analogized to a configuration of springs with one spring placed along each surface edge  304  not on surface boundary  308 . Surface boundary  308  is treated like a rigid framework within which surface vertices  306  are then allowed to relax, forming map vertices  406  in their rest position (i.e., the position of minimum energy). 
     Referring to example 3-D surface  300  in FIG. 3B, and example planar map  400  in FIG. 4A, a conventional relaxation technique is applied to surface vertices 5, 6, 7, and 8 to form the corresponding map vertices 5, 6, 7, and 8. FIG. 4A depicts a likely result of applying a conventional relaxation technique to surface vertices 5, 6, 7, and 8. 
     FIG. 9 is a flowchart  900  which illustrates step  506  in further detail. Once surface boundary  308  has been mapped to map boundary  408  in step  502 , a relaxation technique is applied to those surface vertices not on the surface boundary in step  902  (as described above), forming those map vertices  406  not on map boundary  408 . 
     According to a preferred embodiment, in step  904  user  102  has the option of performing a “pinning” operation. If user  102  desires to perform a pinning operation, program flow continues to step  906 . Otherwise, program flow returns to step  204  in FIG. 2, where user  102  is once again able to define surface boundary  308 . 
     In step  906 , user  102  selects one or more map vertices and pins them to a particular location(s). Pinning refers to modifying the position (in the 2-D plane in which the planar map is defined) of the selected map vertices  406  and fixing the new position so that when the relaxation technique is again applied in step  902 , the “pinned” map vertices are held in place. As a result, those map vertices which are pinned, like those which are on map boundary  408 , are treated like a rigid framework and the remainder of the map vertices are allowed to relax. 
     FIG. 10A depicts an example planar map  1000  which will be used to illustrate the pinning operation. For purposes of this example, assume that a surface boundary (not shown) defined on a 3-D surface (not shown) was mapped to map boundary  408  (as shown in FIG. 10A) in step  502 . Assume further that a relaxation technique was applied in step  902  to those surface vertices not on the surface boundary, forming map vertices 5, 6, 7, and 8. Those skilled in the art will recognize that, for many purposes, map primitive 1-5-8-4 might be of little use given its small size. For example, assume that planar map  1000  represents a texture map in a computer graphics system, and that map primitive 1-5-8-4 is too small to be useful and is a distorted representation of the corresponding surface primitive 1-5-8-4 (not shown). User  102  may manually correct for these problems using pinning. 
     According to a preferred embodiment, user  102  uses a pointing device (e.g., mouse, joystick, trackball) available with computer system  104  to “drag” map vertex 8 to a new position, such that the redefined map primitive 1-5-8-4 is less distorted and has an increased area. As shown in FIG. 10B, user  102  has dragged map vertex 8 to a new position. Now, map primitive 1-5-8-4 has a larger area and, presumably, is more representative of surface primitive 1-5-8-4 (i.e., less distortion). 
     Map vertex 8 has now been pinned to the new location shown in FIG.  10 B. When program flow again returns to step  902  (after all pinning is complete), a relaxation technique is applied to map vertices 5, 6, and 7, which will likely relax to a new configuration (depending on the particular configuration and relaxation technique used). Map vertices 1,2,3, and 4 are held rigid because they are on map boundary  408 , and map vertex 8 is held rigid because it has been pinned. 
     Program flow continues back to step  904 , where user  102  again has the option to perform additional pinning. In this manner, user  102  may continue altering planar map  1000  until a desired result is achieved. Those skilled in the art will recognize the advantages of this approach, particularly given extremely complex planar maps with thousands of map vertices. Often, it is desirable to iteratively modify these complex planar maps by pinning key map vertices and relaxing the rest, until each map primitive has an acceptable shape and size. 
     Map Boundary Interpolation 
     The edge-and-angle proportional mapping method described above can generate planar maps having undesirable qualities. For example, the aforementioned methods can generate a map boundary which crosses itself (“self-crossing” boundary). Self-crossing map boundaries are undesirable because it is no longer true that all points on a 3-D surface map to a unique point on the planar map-the self-crossing map boundary creates an overlap region where a single point on the planar map corresponds to two or more points on the 3-D surface. Another example of an undesirable planar map is one that contains areas which are unacceptably distorted. Those skilled in the art will recognize other instances where the aforementioned methods produce undesirable planar maps. 
     Returning to FIG. 5, steps  504  and  508  describe an extension to the mapping methods described above whereby the map boundary generated in step  502  is interpolated with a different map boundary generated according to a conventional boundary mapping technique. This interpolation can alleviate some of the undesirable effects mentioned above. 
     Alternatively, the interpolation described in steps  504  and  508  can be used even where the aforementioned methods generated an acceptable planar map to provide an alternate, possibly superior in some respect(s), planar map. 
     For purposes of illustration, assume that step  502  has created a self-crossing map boundary, as depicted in FIG. 11A, based on a 3-D surface (not shown) with a defined surface boundary. Map edge 1s-2s crosses map edge 5s-6s and map edge 2s-3s crosses map edge 4s-5s. This map boundary will be referred to as the source map boundary  1102 . 
     In step  504 , program flow continues on to step  508  where interpolation is to be performed, or to step  506  where the interpolation is not to be performed. In a preferred embodiment, user  102  determines whether or not map boundary interpolation will be performed. Those skilled in the art will recognize that this determination could also be made automatically. For example, in an alternate embodiment interpolation is automatically performed whenever a self-crossing boundary is detected (according to conventional numerical techniques). 
     In those instances where map boundary interpolation is to be performed, program flow proceeds to step  508 . FIG. 12 is a flowchart illustrating step  508  in greater detail. In step  1202 , another planar map is generated according to a conventional boundary mapping method. 
     Any conventional boundary mapping method may be used in step  1202 . For example, an edge proportional boundary mapping method can be used. Edge proportional methods map the surfaces vertices on the surface boundary to the perimeter of a pre-selected geometric shape in proportion to the length of the surface edges forming the surface boundary. Edge proportional methods produce map boundaries which are convex, a property which is particularly useful, for example, when the interpolation is used to correct for a self-crossing map boundary generated in step  502 , as will be described below. Convex boundaries, by definition, do not self-cross. 
     For purposes of illustration, assume that an edge proportional method is used in step  1202 . FIG. 11B depicts an example map boundary generated by applying an edge proportional method to the same 3-D surface used to generate source map boundary  1102 . In a preferred embodiment, the pre-selected geometric shape is a rectangle  1112 , though other shapes may be used. All of the surface vertices on the surface boundary are mapped onto rectangle  1112 . The distance between them, measured along the rectangle perimeter, is proportional to the length of the surface edge connecting the vertices-the resulting map boundary is therefore “edge proportional.” 
     These distances are calculated by multiplying the total length of the rectangle perimeter by the length of the surface edge, and dividing that product by the total length of the surface boundary. Once map vertices  1108  are placed, they are connected to form map edges  1110 . For example, in FIG. 11B map vertices  1108  are first placed along rectangle  1112 , where the distances between map vertices  1108  are calculated as described above. Once map vertices  1108  are placed, map edges  1110  are created in “connect-the-dots” fashion. This map boundary will be referred to as the target map boundary  1104 . 
     In step  1204 , source map boundary  1102  and target map boundary  1104  are interpolated to form a new map boundary. According to a preferred embodiment, the interpolation is computed as follows: 
     
       
           new   —   angle   i   =source   —   angle   i (1−α)+ target   —   angle   i *α 
       
     
     
       
           new   —   length   i   =source   —   length   i (1−α)+ target   —   length   i *α 
       
     
     The interpolation parameter α controls the relative weight accorded source map boundary  1102  and target map boundary  1104 . The source and target map boundaries have the same number of map vertices (and map edges) because they were generated using the 3-D surface and surface boundary. Referring to FIGS. 11A and 11B, source map boundary  1102  has six map vertices 1s to 6s and target map boundary  1104  map vertices 1t to 6t(“s” and “t” designate source and target). 
     FIG. 11C depicts an interpolated map boundary  1110 , which is interpolation of source map boundary  1102  and target map boundary  1104  according to the formula above. The planar angle at each map vertex on the interpolated map boundary is equal to the weighted average of the planar angles at the corresponding map vertices on the source and target map boundaries, where the relative weighting is determined by interpolation parameter α. For example, the planar angle at map vertex in (new_angle 1 ) is calculated by multiplying (1−α) times the planar angle at map vertex Is (source_angle 1 ) and adding a times the planar angle at map vertex It (target_angle 1 ). Similarly, the length of map edge 1n-2n (new_length 1 ) is calculated by multiplying (1−α) times the length of map edge 1s-2s (source length 1 ) and adding a times the length of map edge 1t-2t (target_length 1 ). 
     As shown in FIG. 11C, interpolated map boundary  1106  is no longer a self-crossing boundary. However, this result is not guaranteed in every case, depending on the particular boundaries involved and the value chosen for the interpolation parameter α. If α is chosen such that the source map boundary is too heavily weighted (small α), the resulting interpolated map boundary may still be self-crossing. A non-self-crossing map boundary can eventually be achieved by increasing the value of α, thereby increasing the weight given to the target map boundary. 
     The interpolation parameter α is preferably selected by user  102 . However, those skilled in the art will recognize that a could be pre-set (ie., 0.5, which accords equal weight to the source and target map boundaries) or chosen automatically based on any one of a variety of criteria. 
     This preferred interpolation is easy to implement and requires very little processing time. However, those skilled in the art will recognize that more sophisticated interpolations could be used at the cost of additional complexity and processing time. The fundamental idea remains the same: interpolating a source map boundary generated by an edge-and-angle proportional method with a target map boundary generated by any other conventional method. 
     Referring to FIG. 11C, interpolated map boundary  1110  is not closed at map vertex 6n. The preferred interpolation described above does not guarantee that the interpolated map boundary will be closed-in general the interpolated map boundary will not be closed. Therefore, in step  1206  the interpolated map boundary is closed by displacing the map vertices in the same manner as described above with respect to step  706  in FIG.  7 . 
     Conclusion 
     While various embodiments of the present invention have been described above, it should be understood that they have been presented by way of example only, and not limitation. Thus, the breadth and scope of the present invention should not be limited by any of the above-described exemplary embodiments, but should be defined only in accordance with the following claims and their equivalents. 
     The previous description of the preferred embodiments is provided to enable any person skilled in the art to make or use the present invention. While the invention has been particularly shown and described with reference to preferred embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention.