Abstract:
Deforming a three-dimensional computer-generated model to cause a change of shape of the three-dimensional model includes representing a surface of the model using a surface representation initially comprised of an original surface definition, deriving smooth three-dimensional mapping functions where each mapping function defines a deformation to the surface and at least one mapping function is non-affine, constructing a composition of the mapping functions and the original surface definition where each mapping function is included in the composition in succession in accordance with the order of derivation, and applying the composition after each successive mapping function is included in the composition causing the surface of the three-dimensional model to be deformed while preserving the smoothness to the lowest degree of smoothness of the mapping functions.

Description:
BACKGROUND OF THE INVENTION  
       [0001]     Computer-aided design (CAD) software allows a user to construct and manipulate complex three-dimensional (3D) models. A number of different modeling techniques can be used to create a 3D model. These techniques include solid modeling, wire-frame modeling, and surface modeling. Solid modeling techniques provide for topological 3D models, where the 3D model is a collection of interconnected topological entities (e.g., vertices, edges, and faces). The topological entities have corresponding supporting geometrical entities (e.g., points, trimmed curves, and trimmed surfaces). The trimmed surfaces correspond to the topological faces bounded by the edges. Wire-frame modeling techniques, on the other hand, can be used to represent a model as a collection of simple 3D lines, whereas surface modeling can be used to represent a model as a collection of exterior surfaces. CAD systems may combine these and other modeling techniques, such as parametric modeling techniques. Parametric modeling techniques can be used to define various parameters for different features and components of a model, and to define relationships between those features and components based on relationships between the various parameters.  
         [0002]     Surfaces are represented in CAD systems using several techniques. The techniques include the definition of surfaces as parametric surfaces, mesh surfaces, subdivision surfaces, and implicit surfaces.  
         [0003]     A parametric surface is a deformation of a parameter space, typically that space is rectangular. Computer-aided modeling software commonly uses low-degree algebraic deformations, resulting in various spline and quadric surfaces. For some types of parametric surfaces, a set of control points defines the surface shape.  
         [0004]     A mesh surface, also referred to as a tessellated surface, is a connected set of polygons, which are often triangular polygons. A mesh surface cannot represent a smooth surface exactly. However, certain basic operations applied to mesh surfaces are easy to implement and efficient to execute. Such basic operations include computing a bounding box and intersecting a mesh surface with one or more simple objects.  
         [0005]     A subdivision surface is defined as the limit of a sequence of meshes. Each mesh in such a sequence is obtained by subdividing the previous mesh in the sequence according to one of several subdivision methods, which are know by those who are skilled in the art. An initial control mesh (i.e., the first mesh in the sequence) is thus sufficient to specify the subdivision surface. Although the initial control mesh is not smooth, the resulting subdivision surface is smooth, and the degree of smoothness of the resulting subdivision surface depends on the subdivision method chosen. If desired, the subdivision method can be adjusted to ensure that certain edges remain sharp.  
         [0006]     An implicit surface is defined as the set of points that satisfy the expression F(x,y,z)=0. If F is a smooth function, the surface will usually be smooth. Spheres, tori, and ellipsoids are some of the surfaces that can be represented as implicit surfaces.  
         [0007]     In general, state-of-the-art computer graphics hardware is optimized to render mesh surface representations, and therefore, some surface representations are routinely converted to mesh surfaces for rendering. Approximating a parametric surface with a mesh is generally straightforward because a mesh can be constructed on the parametric space and a deformation of the surface can be evaluated on the mesh vertices. Converting a subdivision surface to a mesh is also generally straightforward, since the sequence of meshes converges to the subdivision surface very quickly. Approximating an implicit surface with a mesh is generally difficult; however, methods have been developed to approximate an implicit surface.  
         [0008]     To deform a spline or a subdivision surface, the surface control mesh is modified. The coarser the control mesh is, the more limited is the ability to control modifications to the surface. However, a finer control mesh (which may be necessary to produce small features), increases the size of the control mesh, and therefore, requires more memory to store the surface and a greater number of computations to transform the surface.  
         [0009]     Another deformation approach is to deform the space in which the surface is located. Typically, the 3D space is tessellated into a cubic mesh and the cubic mesh is deformed. Based on the deformed mesh, a deformation is computed for the entire space and applied to the surface, as described in an article titled “A Framework for Geometric Warps and Deformations” by Tim Milliron et al., ACM Transactions on Graphics, Vol. 21, No. 1, January 2002, pp. 20-51. The deformation may be constrained to only affect a portion of the entire space, as described in an article titled “General Constrained Deformations Based on Generalized Metaballs” by Xiaogang Jin et al., Pacific Conference on Computer Graphics and Applications 1998, pp. 115-124.  
         [0010]     One drawback of defining a space deformation is that deforming the space in which the surface is located is not intuitive because the objective is to deform the surface not the space in which the surface resides. Additionally, if the space deformation is applied to vertices of a mesh, sharp edges of the mesh may be exaggerated because the angles between adjacent polygons may increase. Applying the space deformation to the mesh polygons may require reapproximating the result by another mesh. Reapproximating the result with a refinement of the original mesh often leads to a deterioration of the polygon aspect ratios. Furthermore, if the space deformation is applied to a spline, converting the resulting surface back into a spline may be computationally costly and the smoothness of the surface may not be maintained. Generally, the smoothness constraints between surface patches can be maintained only approximately.  
         [0011]     Many commercially available 3D CAD systems have free-form surface manipulation capabilities. Such capabilities are usually constrained by representing the surfaces internally as non-uniform rational b-spline (NURBS) surfaces. Thus, manipulation controls are often restricted to particular points or curves of the surfaces. The user, therefore, is limited in his or her ability to form a desired shape. Some systems can allow an arbitrary point or curve as a manipulation control. For example, the SolidWorks® 2004 software product, available from SolidWorks Corporation of Concord, Mass., enables one or more surfaces to be deformed using an arbitrary point or curve as a manipulation control. Then, resulting surfaces are approximated with NURBS surface patches. Achieving the desired accuracy of shape and smoothness between surface patches, in general, leads to high density control grids and makes such surfaces computationally expensive for subsequent operations.  
         [0012]     Many users of 3D CAD systems are skilled in 3D modeling techniques and design 3D models professionally. These experienced users understand how to manipulate control points and meshes used to define a surface, and how to manipulate vertices and edges that are surface entities. A 3D CAD system that provides a simple and intuitive means to create and modify the shape of 3D models of various complexities would be advantageous for those not skilled in 3D modeling techniques as well as for those that are. A 3D CAD system that allows users to interactively deform complex models by direct manipulation of an arbitrary point, curve, or surface area, while preserving the smoothness of the surface throughout the model, could enable users to create smooth shapes in a natural, fast, and simple way, and thereby enhance the capabilities and ease of use of a 3D CAD system.  
       SUMMARY OF THE INVENTION  
       [0013]     In general, in one aspect, the invention features a computer-implemented method for deforming a three-dimensional computer-generated model. The method includes representing a surface of the three-dimensional model using a surface representation that includes an original surface definition. Smooth three-dimensional mapping functions that define a deformation to the surface are derived, where at least one of the three-dimensional mapping functions is non-affine. A composition of the three-dimensional mapping functions and the original surface definition is constructed, where each of the mapping functions is included in the composition in succession. Moreover, the mapping functions are ordered in the composition in the order in which the mapping functions were derived. The composition is applied after each successive three-dimensional mapping function is included in the composition, causing the surface of the three-dimensional model to be deformed while preserving smoothness to the lowest degree of smoothness of the mapping functions.  
         [0014]     Implementations may include defining the surface as a parametric surface, a mesh surface, or a subdivision surface, and using the chain rule to determine a derivative of the surface representation.  
         [0015]     A user interface that includes depictions of deformation results can be displayed by some implementations and one depiction can be selected to indicate a deformation operation that is implemented as a mapping function. The deformation operation can allow for a curve, a point, or an area of the surface to be pulled. Additionally, the deformation operation can impress a sphere or a curve upon the surface, or narrow, flatten, or twist the surface.  
         [0016]     Implementations may also include one or more of the following features. A deformation control that is positioned with respect to the surface of the model can be identified, transformed, and utilized as a parameter to the mapping function. The deformation control can be a lower-order geometry, and can be a point, a curve, a plane, a line, an axis, an area of a surface, or a parabolic object.  
         [0017]     The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description that follows. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. 
     
    
     DESCRIPTION OF THE DRAWINGS  
       [0018]      FIG. 1  is an illustration of a computer-generated model displayed in a window.  
         [0019]      FIG. 2  is an illustration of a set of icons.  
         [0020]      FIG. 3  is an illustration of a deformed 3D model.  
         [0021]      FIG. 4  is an illustration of a deformed 3D model.  
         [0022]      FIG. 5A  is an illustration of a curve on a 3D model.  
         [0023]      FIG. 5B  is an illustration of a deformed 3D model.  
         [0024]      FIG. 6A  is an illustration of a deformed 3D model.  
         [0025]      FIG. 6B  is an illustration of a deformed 3D model.  
         [0026]      FIG. 7  is a flowchart of a procedure for directly manipulating a 3D model.  
         [0027]      FIG. 8  is an illustration of a data structure.  
         [0028]      FIG. 9  is a diagram of a computer system.  
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0029]     The present invention facilitates smooth real-time deformations of freeform surfaces by utilizing arbitrary lower-order geometry and other deformation controls to directly manipulate one or more surfaces. Such arbitrary lower-order geometry includes, by way of non-limiting example, points, curves, planes, axes, and surfaces. Moreover, the present invention preserves the existing smoothness of the surfaces and existing surface curvature characteristics. A sequence of smooth space mappings is used to change the shape of a model without a loss in model accuracy. Furthermore, a user interface provides users with a natural, fast, and simple approach to deforming freeform surfaces. The user interface affords access to a functional library that supports the direct interactive manipulation of one or more surfaces. Due to the foregoing, the present invention enhances the capabilities of a computerized modeling system.  
         [0030]      FIG. 1  shows a window  102  displayed on a CRT and generated by modeling software executed by a computerized modeling system, an example of which is later shown with reference to  FIG. 9 . The window  102  is a conventional computer-generated window that can be programmed by one of ordinary skill in the art using conventional, commercially available, software programming tools, such as those available from Microsoft Corporation of Redmond, Wash.  
         [0031]     A computer-generated 3D model  104  is displayed within a modeling portion  106  of the window  102 . The surfaces of the 3D model  104  can be displayed, or the 3D model  104  can be displayed using solid lines and dashed lines to show visible edges and hidden edges, respectively, of the 3D model. Implementations also may include other window areas. A set of file icons  108  aids the user in the selection of various files that may be used as a base model, such as a sphere primitive. A set of deformation icons  110  enables a user to manipulate the model  104  shown in the modeling portion  106  of the window  102 .  
         [0032]      FIG. 2  shows the set of deformation icons  110 . Each of the deformation icons  110  depicts a deformation result and when selected indicates to the modeling software that a particular function is to be applied to a local region of a model. Three such functions enable a point on a surface, a curve on a surface, and an area of a surface to be pulled, and are activated by selecting a point pull icon  212 , curve pull icon  214 , and an area pull icon  220 , respectively. When a curve, a point, or an area is pulled the surface is modified in real time. Thus, the pulling action and resulting modification caused by the pulling action is interactive. Another function enables a spherical indentation to be impressed upon a surface and is activated when a poke sphere icon  216  and a center point for the sphere with respect to a model are selected. Still another function causes a surface to narrow at a specific location and is activated when a create neck icon  224  is selected, after which a plane is displayed and positioned such that the narrowing occurs in a direction towards the plane. An arbitrary point on the plane is also indicated to designate the center of the narrowing affect. A flatten surface icon  218 , when selected, causes a surface to be flattened after a plane is positioned where the surface is to be flattened. A curve imprint icon  222 , when selected, causes a curve to be impressed upon a surface. As is the case with the pulling functions, all other functions are applied to a model in real time, and therefore, the deformations that result occur interactively.  
         [0033]      FIG. 3  is an illustration of a 3D model  330  deformed by directly manipulating an arbitrary point  332  (attached to a cursor object  334 ) positioned on the circumference of what was initially a sphere. The initial sphere can be displayed after selecting file icon  336 . To deform the 3D model  330  as is shown in  FIG. 3 , a user first selects the point pull deformation icon  212  then selects the arbitrary point  332 . The user can interactively move the arbitrary point  332  using a pointing device (e.g., a mouse). In response to the movement of the arbitrary point  332 , the 3D model  330  is deformed in real time.  
         [0034]      FIG. 4  shows a smooth deformation of the 3D model  330  after the direct manipulation of the arbitrary point  332 , which is attached to the cursor object  334 . In  FIG. 4 , the 3D model  330  takes on a different appearance than that in  FIG. 3  because a larger area on the surface of the 3D model  330  is influenced by the displacement of the arbitrary point  332 . A user interface control, slider  434 , is used to increase and decrease the area of influence. When the slider  434  is moved toward the bottom of the window  102 , a larger portion of the model is subsumed in the area of influence; whereas, when the slider  434  is moved toward the top of the window  102 , a smaller portion of the model is subsumed in the area of influence.  
         [0035]      FIGS. 5A and 5B  illustrate the deformation of a 3D model  502  by manipulating an arbitrary curve  504  drawn on the 3D model  502 . As is the case when a 3D model is deformed via direct manipulation of an arbitrary point, the deformation that results from the direct manipulation of an arbitrary curve is a localized deformation. A curve can be drawn on a model after the curve symbol  506  is chosen from the user interface. Other symbols  508  may be chosen from the user interface that enable a straight line and an arc to be drawn on the model and then be used to directly manipulate the model.  FIG. 5B  illustrates the 3D model  502  after the curve  504  is pulled using a pointing device. As shown in  FIG. 5B , the transformation of the curve  504  extrudes the surface. The orientation of the 3D model  502  differs in  FIGS. 5A and 5B  because the curve  504  used to deform the 3D model  502  and the deformation of the 3D model  502  are viewed more advantageously from different viewpoints.  
         [0036]      FIGS. 6A and 6B  illustrate global deformations being applied to model  104 . In  FIG. 6A , the global deformation is the result of an application of a scale function. The scale function is activated after the scale icon  606  is selected from the user interface, after which a three-dimensional enclosure is drawn around the model  104 . A corner or a face of the enclosure can be moved to change the size or the shape of the enclosure and thereby cause a scale function to be applied to the model  104 . Conic handles  608  are provided to control the movement of the enclosure.  
         [0037]     Other functions that have a global affect on a model include a bend function and a twist function, which are activated when a bend icon  612  and a twist icon  614  are selected, respectively.  FIG. 6B  shows model  104  after a bend function has been applied to the model  104 . After the bend icon  612  is selected, a parabolic object  616  is displayed in the modeling area  106 . A pivot point  618  is designated on the parabolic object  616 , as are leverage points  620 . The pivot point  618  and the leverage points  620  can be moved using a pointing device. As the pivot point  618  or one of the the leverage points  620  are moved, the parabolic object  616  bends and the model  104  is deformed by bending in the same manner as the parabolic object  616 . When the twist icon  614  is selected, a rotational control and an axis is displayed to control the twisting motion.  
         [0038]     Referring now to  FIG. 7 , a procedure  700  for directly manipulating a 3D model is shown. First, a representation of the model is defined (step  702 ). The model representation may include any surface representations that can be evaluated, including parametric surfaces, mesh surfaces, and subdivision surfaces. In general, the present invention applies to any 3D representation of a model that can be evaluated.  
         [0039]     In the next step, a deformation operation is chosen and a deformation control is identified (step  704 ). The deformation operation may be chosen by selecting one of the deformation icons  110  or one of the icons depicting global deformation operations, which were previously discussed with reference to  FIG. 2 , and  FIGS. 6A and 6B , respectively. For example, the curve pull icon  214  shown in  FIG. 2  can be selected to choose a deformation operation that deforms the model by applying a shape of a curve to the model, or the poke sphere icon  216  (also shown in  FIG. 2 ) can be selected to choose a deformation operation that causes a spherical impression on one or more surfaces of the model. The deformation control can be any lower-order geometry. By way of non-limiting example, the identified deformation control may be a point, a curve, a plane, an axis, or an area of a surface, and can be identified by selecting a point, drawing a curve, positioning a plane, or specifying a surface region. The deformation control may also be an object used to apply a global deformation. In any case, the deformation control complements the chosen deformation operation. For example, pull operations can be controlled by a point, a curve, or an area of a surface; scale operations can be controlled by the conic handles of a rectangular frame (as shown in  FIG. 6A ); bend operations can be controlled by a pivot point and leverage points (as shown in  FIG. 6B ). In addition, some deformation controls may be implied by virtue of the nature of the chosen deformation operation.  
         [0040]     In the next step, one or more additional constraints may be defined (step  706 ). One such additional constraint is a region of influence. When a region of influence is defined as a constraint, only the points within an area that surrounds the deformation control are affected by the deformation operation. By way of non-limiting example, the region of influence may be a three-dimensional region specified by a radius of a specified size surrounding a deformation control. Moreover, the size of the region of influence may be a system-defined size or a user-defined size. Another constraint may be an area constraint, and in this case, only a selected area will be affected by the deformation operation. The selected area can be indicated by drawing a contour that surrounds an area to be affected by a deformation.  
         [0041]     Yet another constraint is a pattern constraint. A pattern constraint, when defined, replicates a deformation in a way required by a pattern, such as a rectangular, a circular, or a mirror pattern. In the case of a mirror pattern constraint, a mirror icon is chosen from the user interface causing a selected component to be symmetrically duplicated on an opposite side of a plane positioned by a user. Then, when one of the symmetrical components is deformed, a symmetrical deformation is applied to the other symmetrical component simultaneously.  
         [0042]     Other additional constraints may include a half-space constraint or a space-band constraint. A half-space constraint limits the deformation operations to points positioned on one side of a specified plane. A space-band constraint limits the deformation operations to points positioned between two parallel planes.  
         [0043]     In the next step, the deformation control is moved or modified (step  708 ). The deformation control may be moved by means of customary user interactions using a pointing device, such as a mouse, to drag the deformation control (e.g., a point, a curve, or an area of a surface) to a new location in the modeling area. The deformation control can also be modified. By way of non-limiting example, a deformation control that is initially a straight line can become curved, a deformation control that is a curve can change shape (e.g., bend), or a deformation control that is a boundary enclosing a region can expand, contract, or be modified in another way (e.g., skewed).  
         [0044]     In some cases when a deformation control is moved or modified, a deformation target is produced. The deformation target represents the deformation control in a new location or represents the deformation control assuming a new shape. The deformation target is defined using the same representation as the deformation control. That is, if the deformation control is a point or a curve, the deformation target is represented as point or a curve, respectively.  
         [0045]     The procedure  700  then derives a 3D mapping function from a functional library (step  710 ). The 3D mapping function, when applied to the 3D model, maps the deformation control to the deformation target, if applicable, while considering the additional constraints defined in step  706 . If the 3D mapping function is smooth, the 3D mapping function automatically preserves the degree of smoothness in and between the surfaces in the model. For example, C 2  mapping preserves C 2  continuity within and between surfaces.  
         [0046]     A series of 3D mapping functions derived in step  710 , one per each cycle of loop  720 , determine the deformed shape of the one or more surfaces being manipulated. The internal representation of the model is maintained as a collection of warp surfaces, where a warp surface as implemented by the present invention is a sequence of smooth 3D mapping functions applied to the original surface definition. Rather than have each deformation redefine the initial definition of a surface of the model, a sequence of smooth 3D mapping functions and an original surface definition are stored and associated with one another. Each smooth 3D mapping function derived in step  710  per cycle of loop  720  represents one deformation in the sequence.  
         [0047]     One advantage of utilizing warp surfaces is that any parametric surface can be used as the original surface without conversion. Applying the same sequence of 3D mappings to a collection of parametric surfaces, such as a B-rep surface (i.e., a boundary-representation surface), typically preserves both the topology of the collection of parametric surfaces and the continuity relationships between the parametric faces. Another advantage of utilizing warp surfaces is that no approximation is necessary for the operations, and therefore, precision is maintained as the surface is manipulated. Implementations may benefit from display and query optimizations by using a mesh approximation to the warp surface; however, this approximation is not incorporated into the surface representation. Another advantage of utilizing warp surfaces is that because a precise evaluator (i.e., a method for computing a value of a function and the derivatives thereof) for the surface is available, warp surfaces have all of the advantages of parametric surfaces (e.g., the existence of methods for approximating the surface with a mesh and for various geometric operations). A further advantage of warp surfaces is that relatively little computation is necessary to update the model state with a new deformation, so the model can be manipulated interactively. Another advantage of warp surfaces is that each deformation can be described with a small number of parameters, resulting in a concise representation for the entire model with regards to disk storage space.  
         [0048]     In the next step of procedure  700 , the 3D mapping function is applied to the current model state (step  712 ), where the current model state is the original representation of the model having all transformations (i.e., 3D mapping functions) applied thus far. For example, if the model representation is defined as a polygonal mesh, the model state is produced by applying the sequence of transformations to the vertices of the polygonal mesh. To increase the accuracy of the mesh approximation of the deformed model, in the areas of higher curvature, the density of the polygonal mesh can be increased automatically.  
         [0049]     The procedure  700  then determines whether to continue to deform the model using the identified deformation control, selected deformation operation, and defined constraints (step  714 ). The determination to continue may occur when further input is received to move or modify the deformation control. In the case when the determination to continue occurs, the most recent 3D mapping operation derived will be replaced by a subsequently derived 3D mapping function.  
         [0050]     When the deformation to the model using the identified deformation control, chosen deformation operation, and defined constraints is complete, the procedure  700  then determines whether to apply a new feature to the model (step  716 ). If a new feature is to be applied to the model, another deformation operation may be chosen and another deformation control may be identified (step  704 ), and/or additional constraints may be defined (step  706 ). The deformation control is then moved or modified (step  708 ). Then, a new 3D mapping function is derived (step  710 ) and applied to the current model state (step  712 ), which now includes the previously derived 3D mapping function. The new 3D mapping function will become another transformation in the sequence of transformations used to re-define the model and will be replaced by a subsequently derived 3D mapping functions until the determination is made not to continue to deform the model using the then identified deformation control, selected deformation operation, and defined constraints. When the procedure  700  determines that no new features will be applied to the model (in step  716 ), procedure  700  completes (step  718 ).  
         [0051]     In one embodiment, the deformed model can be presented to the user for preview purposes. In such an embodiment, a preview step occurs immediately after a determination is made not to continue to deform the model (i.e., after step  714  and prior to step  716 ). The preview step displays a user interface dialog box that queries the user whether the deformation should be accepted. The user then has the opportunity to accept the deformation by pressing an accept button in the dialog box, in which case, the last 3D mapping function that was derived is added to the warp surface sequence of 3D mapping functions. If the user presses a reject button displayed in the dialog box, the last 3D mapping function that was derived is not added to the warp surface sequence of 3D mapping functions.  
         [0052]     Referring now to  FIG. 8 , an illustration of a data structure  800  of a model that may be used by the present invention is shown. The data structure  800  is an abstract representation of a model defined by three warp surfaces  830 ,  840 , and  850 . The three surfaces are stored as surface  1  definition  832 , surface  2  definition  842 , and surface  3  definition  852 . Three 3D mapping functions, which were derived in step  710  of  FIG. 7 , apply to one or more of the three surfaces. One 3D mapping function applies to surface  1  definition  832  and surface  2  definition  842 , and is stored in the data structure  800  twice as first 3D mapping function  834  and first 3D mapping function  844 . A second 3D mapping function  836  only applies to surface  1  definition  832  and is stored in the data structure  800  associated with surface  1  definition  832 . A third 3D mapping function applies to surface  1  definition  832 , surface  2  definition  842 , and surface  3  definition  852 , and is stored in the data structure  800  as third 3D mapping function  838 , third 3D mapping function  848 , and third 3D mapping function  858 .  
         [0053]     The 3D mapping functions that are derived in step  710  and illustrated in  FIG. 8  define a particular deformation to be applied. Those skilled in the art can design a variety of mapping functions that can define the deformations discussed herein, as well as other deformations. In general, designing freeform deformations is an under-constrained problem, meaning that a large number of possible deformations can have the desired properties. Following is a description of some of the possible definitions of the deformations heretofore discussed, and are by way of non-limiting example. The definitions are expressed as functions of {overscore (x)}, where {overscore (x)} is a point in 3D space.  
         [0054]     In one embodiment, as the deformation control is moved or modified, the points closest to the deformation control have the greatest movement. That is, the points near the outer boundary of the region of influence move by a lesser amount than the points closer to the deformation control, and points on and beyond the outer boundary of the region of influence do not move at all.  
         [0055]     In the following two examples, the function denoted h(t) is used to build a smooth transition from the region near the control where the deformation is maximal to the outer boundary of the region of influence. The function is C 2  on its entire domain and is defined as follows:  
         h   ⁡     (   t   )       =     {           0   ,             if   ⁢           ⁢        t          &gt;   1                   (     1   -     t   2       )     3     ,             if   ⁢           ⁢        t          ≤   1                 
 
         [0056]     A mapping function designed for a point pull operation can be expressed as follows:  
         f   ⁡     (     x   _     )       =     {         h   ⁡     (              x   _     -     a   _            R     )       ·     (       B   _     -     a   _       )       +     x   _             
 
 where 
        {overscore (a)} is the position of the deformation control,     {overscore (B)} is the target point, and     R is the radius of influence.        
 
         [0060]     An example of a mapping function that is implemented in an embodiment to produce a narrowing of a modeled object (i.e., create a neck) can be expressed as follows:  
         f   ⁡     (     x   _     )       =     {       x   _     -       (         x   _     -     c   _       ,     n   _       )     ·     n   _       +       (     1   +       (     s   -   1     )     ·     h   ⁡     (       (         x   _     -     c   _       ,     n   _       )     R     )           )     ·     (         x   _     -     c   _       ,     n   _       )       -     n   _             
 
 where 
        {overscore (c)} is the center point of the neck feature,     s is a scale factor, and s&gt;0,     R is the radius of influence,     {overscore (n)} is the unit vector normal to the plane positioned through the center point and such that points of a modeled object are pulled toward the center point along the plane, and     ({overscore (x)}−{overscore (c)}, {overscore (n)}) denotes the dot product of the vectors {overscore (x)}−{overscore (c)} and {overscore (n)}.        
 
         [0066]     In an embodiment, the mapping function for impressing a spherical feature upon a modeled object can be expressed as follows:  
         f   ⁡     (     x   _     )       =       c   _     +       (     R   +     r   ·     g   ⁡     (                x   _     -     c   _            -   R   +   r       2   ⁢   r       )           )     ·         x   _     -     c   _                x   _     -     c   _                      
 
 where 
        {overscore (c)} is the center of the sphere acting upon the model,     R is the radius of the sphere,     r is the radius of transition between the impressed sphere and the edge of the radius of influence, and  
         g   ⁡     (   t   )       =     {           0   ,             if   ⁢           ⁢   t     &lt;   0                   t   3     ⁡     (     2   -   t     )       ,             if   ⁢           ⁢   0     ≤   t   ≤   1                   2   ⁢   t     -   1     ,             if   ⁢           ⁢   t     &gt;   1                 
       
 
         [0070]     The foregoing expressions of 3D mapping functions are C 2  continuous in space, and therefore, C 2  continuity of the surface is preserved. Additionally, the foregoing expressions of 3D mapping functions are non-affine 3D mapping functions. Furthermore, the foregoing expressions provide natural-looking shape deformations and permit efficient implementations.  
         [0071]     From a user perspective, the deformations described herein are intuitive because each deformation is specified as a deformation to a warp surface while internally the appropriate space deformation (i.e., 3D mapping) is constructed. In one embodiment, the original surface is defined as a parametric surface, s(u,v): [0,1] 2 →R 3 , which may possibly be a rectangle. When a mapping f: R 3 →R 3  is applied to the original surface, the new parametric surface can be expressed as {tilde over (s)}(u,v)=f·s(u,v). After n mappings, f 1 , . . . f n , the new parametric surface is the composition of the mappings and the original parametric surface, that is f n ∘f n-1 ∘ . . . ∘f 1 ∘s(u,v). Although the foregoing describes an embodiment utilizing a parametric surface, the original surface can be defined as any type of surface that can be evaluated.  
         [0072]     To display the parametric surface and perform other computations on the surface efficiently, a mesh is maintained that approximates the surface. When a new deformation is added to the 3D mapping sequence, the new deformation is applied to the vertices of the mesh to achieve a mesh that approximates the deformed surface. If the mesh is too coarse after the deformation, the surface is evaluated at additional points and the mesh is refined where necessary. A mesh can be generated using techniques such as those described in an article titled “Bubble Mesh: Automated Triangular Meshing of Non-Manifold Geometry by Sphere Packing” by Kenji Shimada and David C. Gossard, Third ACM Symposium on Solid Modeling and Applications, ACM Press 1995, pp. 409-419.  
         [0073]     Various modeling operations (e.g., surface intersection, rendering, and mesh refinement), require or may be enhanced by the ability to evaluate first and second derivatives of the surface. In a preferred embodiment, 3D mapping evaluators can handle first and second derivatives. In one embodiment, the chain rule (known by those of ordinary skill in the art for taking the derivative of compositions of functions), is applied to compute exact first and second derivatives of a warp surface at an arbitrary point.  
         [0074]     Additional advantages of the present invention include providing the ability to use arbitrary lower-order geometry to control smooth real-time deformations and the efficient disk storage of the model as one or more warp surfaces. The use of lower-order geometry to manipulate a model offers users of the present invention a natural, fast, and simple approach to deforming freeform surfaces. Whereas, storing the model as one or more warp surfaces is not only efficient in terms of storage space, but also because storage as a warp surface enables the accuracy of the model to be maintained.  
         [0075]      FIG. 9  shows a computerized modeling system  900  that includes a CPU  902 , a CRT  904 , a keyboard input device  906 , a mouse input device  908 , and a storage device  910 . The CPU  902 , CRT  904 , keyboard  906 , mouse  908 , and storage device  910  can include commonly available computer hardware devices. For example, the CPU  902  can include a Pentium-based processor. The mouse  908  may have conventional left and right buttons that the user may press to issue a command to a software program being executed by the CPU  902 . As an alternative or in addition to the mouse  908 , the computerized modeling system  900  can include a pointing device such as a trackball, touch-sensitive pad, or pointing device and buttons built into the keyboard  906 . Those of ordinary skill in the art appreciate that the same results described herein with reference to a mouse device can be achieved using another available pointing device. Other appropriate computer hardware platforms are suitable as will become apparent from the discussion that follows. Such computer hardware platforms are preferably capable of operating the Microsoft Windows NT®, Windows® 98, Windows® 2000, Windows XP®, Windows ME®, UNIX®, Linux®, or MAC OS® operating systems.  
         [0076]     Additional computer processing units and hardware devices (e.g., rapid prototyping, video, and printer devices) may be included in the computerized modeling system  900 . Furthermore, the computerized modeling system  900  may include network hardware and software thereby enabling communication to a hardware platform  912 , and facilitating communication between numerous computer systems that include a CPU and a storage system, among other computer components.  
         [0077]     Computer-aided modeling software may be stored on the storage device  910  and may be loaded into and executed by the CPU  902 . The modeling software allows a user to create and modify a 3D model, and implements aspects of the invention described herein. The CPU  902  uses the CRT  904  to display a 3D model and other aspects thereof as described. Using the keyboard  906  and the mouse  908 , the user can enter and modify data associated with the 3D model. The CPU  902  accepts and processes input from the keyboard  906  and mouse  908 . The CPU  902  processes the input along with the data associated with the 3D model and makes corresponding and appropriate changes to that which is displayed on the CRT  904  as commanded by the modeling software. In one embodiment, the modeling software is based on a solid modeling system that may be used to construct a 3D model consisting of one or more solid and surface bodies. Additionally, the modeling software may allow for relationships that parametrically constrain the definitions of one or more bodies or features with respect to one another. If a parametrically constrained relationship exists between two features, a geometric modification to one feature may cause a geometric modification in the other feature.  
         [0078]     The invention may be implemented in digital electronic circuitry, or in computer hardware, firmware, software, or in combinations thereof. Apparatus of the invention may be implemented in a computer program product tangibly embodied in a machine-readable storage device for execution by a programmable processor; and method steps of the invention may be performed by a programmable processor executing a program of instructions to perform functions of the invention by operating on input data and generating output. The invention may advantageously be implemented in one or more computer programs that are executable on a programmable system including at least one programmable processor coupled to receive data and instructions from, and to transmit data and instructions to, a data storage system, at least one input device, and at least one output device. Each computer program may be implemented in a high-level procedural or object-oriented programming language, or in assembly or machine language if desired; in any case, the language may be a compiled or interpreted language. Suitable processors include, by way of non-limiting example, both general and special purpose microprocessors. Generally, a processor will receive instructions and data from a read-only memory and/or a random access memory. Storage devices suitable for tangibly embodying computer program instructions and data include all forms of non-volatile memory, including by way of example semiconductor memory devices, such as EPROM, EEPROM, and flash memory devices; magnetic disks such as internal hard disks and removable disks; magneto-optical disks; and CD-ROM disks. Any of the foregoing may be supplemented by, or incorporated in, custom-designed ASICs (application-specific integrated circuits).  
         [0079]     A number of embodiments of the present invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. For example, implementations may change the order in which operations are performed. Furthermore, depending on the needs of an implementation, particular operations described herein may be implemented as a combined operation, eliminated, added to, or otherwise rearranged. For example, in  FIG. 7 , the step defining additional constraints (i.e., step  706 ) may be eliminated. Moreover, the deformation control may be identified prior to and independent of choosing a deformation operation and one deformation control may be identified for multiple selections of various deformation operations. Likewise, one deformation operation may be chosen while multiple deformation controls are identified.  
         [0080]     Furthermore, an implementation may allow for more than one deformation operation to be chosen at a time. In such an implementation, a deformation control may be identified for each chosen deformation operation, and each deformation control may be independently moved and modified. This enables the user to simultaneously establish a number of possible interactions by selecting pairs of deformation operations and a deformation controls.  
         [0081]     Accordingly, other embodiments are within the scope of the following claims.