BICUBIC UNIFORM B-SPLINE SURFACE REFINEMENT

Transcription

1 On-Line Geometric Modeling Notes BICUBIC UNIFORM B-SPLINE SURFACE REFINEMENT Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science Uniersity of California, Dais Oeriew Subdiision surfaces are based upon the binary subdiision of the uniform B-spline surface. In general, they are defined by a initial polygonal mesh, along with a subdiision (or refinement) operation which, gien a polygonal mesh, will generate a new mesh that has a greater number of polygonal elements, and is hopefully closer to some resulting surface. By repetitiely applying the subdiision procedure to the initial mesh, we generate a sequence of meshes that (hopefully) conerges to a resulting surface. As it turns out, this is a well known process when the mesh has a rectangular structure and the subdiision procedure is an extension of binary subdiision for uniform B-spline surfaces. Ed Catmull and Jim Clark, while still graduate students at the Uniersity of Utah, sought to extend Doo and Sabin s algorithm to bicubic surfaces. Since the methods of Doo-Sabin is based upon the binary subdiision of the uniform bi-quadratic B-spline patches, Catmull and Clark belieed that study of the cubic case would lead to a better subdiision surface generation scheme. A Matrix Equation for the Bicubic Uniform Spline Surfaces Consider the bicubic uniform B-spline surface P(u, ) defined by the array of control points P 0,0 P 0, P 0, P 0,3 P,0 P, P, P,3 P = P,0 P, P, P,3 P 3,0 P 3, P 3, P 3,3

2 where P(u, ) = u u u 3 ] MP M T 3 where M is the matrix M = The control point mesh for such a patch is shown in the figure below. +,-. /0 0!" # $%& % '() * :;< = >?@ A Subdiiding the Bicubic Patch The bicubic uniform B-spline patch can be subdiided into four subpatches, which can be generated from 5 unique sub-control points. We focus on the subdiision scheme for only one of the four (the subpatch corresponding to 0 u, ), as the others will follow by symmetry. The following figure illustrates the 5 points produced by subdiiding into four subpatches. We hae outlined the initial subpatch that we

3 consider below. It should be noted that the nine interior control points are utilized by each of the four subpatch components of the subdiision. This subpatch can be generated by reparameterizing the surface by the ariables u and, where u = u 3

4 and =. Substituting these into the equation, we obtain P(u, ) = P( u, ) = u ( u ) ( u ) ] 3 MP M T ( ( ) 3 ) = = = = T ] u u u MP M T T ] u u u 3 MM MP M T (M ) T M T ] u u u 3 MSP S T M T u u u 3 ] MP M T where P = SP S T and S = M M

5 Through this process, we hae written the surface P (u, ) as P (u, ) = u u u 3 ] MP M T 3 for some control point array P. This implies that P (u, ) is a uniform bicubic B-spline patch. The matrix S is typically called the splitting matrix, and is straightforward to calculate. It is gien by 0 0 H = By carrying out the algebra, we can calculate the control point array P by P = = 0 0 P 0,0 P 0, P 0, P 0,3 6 0 P,0 P, P, P,3 0 0 P,0 P, P, P,3 0 6 P 3,0 P 3, P 3, P 3, P 0,0 + P,0 P 0, + P, P 0, + P, P 0,3 + P,3 P 0,0 + 6P,0 + P,0 P 0, + 6P, + P, P 0, + 6P, + P, P 0,3 + 6P,3 + P,3 P,0 + P,0 P, + P, P, + P, P,3 + P,3 P,0 + 6P,0 + P 3,0 P, + 6P, + P 3, P, + 6P, + P 3, P,3 + 6P,3 + P 3,3 T

6 and we obtain P 0,0 = P 0,0 + P,0 + P 0, + P, P 0, = P 0,0 + P,0 + 6(P 0, + P, ) + P 0, + P, P 0, = P 0, + P, + P 0, + P, P 0,3 = P 0, + P, + 6(P 0, + P, ) + P 0,3 + P,3 P,0 = P 0,0 + P 0, + 6(P,0 + P, ) + P,0 + P, P, = P 0,0 + 6P,0 + P,0 + 6(P 0, + 6P, + P, ) + P 0, + 6P, + P, 6 P, = P 0, + P 0, + 6(P, + P, ) + P, + P, P,3 = P 0, + 6P, + P, + 6(P 0, + 6P, + P, ) + P 0,3 + 6P,3 + P,3 6 P,0 = P,0 + P,0 + P, + P, P, = P,0 + P,0 + 6(P, + P, ) + P, + P, P, = P, + P, + P, + P, P,3 = P, + P, + 6(P, + P, ) + P,3 + P,3 P 3,0 = P,0 + P, + 6(P,0 + P, ) + P 3,0 + P 3, P 3, = P,0 + 6P,0 + P 3,0 + 6(P, + 6P, + P 3, ) + P, + 6P, + P 3, 6 P 3, = P, + P, + 6(P, + P, ) + P 3, + P 3, P 3,3 = P, + 6P, + P 3, + 6(P, + 6P, + P 3, ) + P,3 + 6P,3 + P 3,3 6 Each of these points can be classified into three categories face points, edge points and ertex points depending on each points relationship to the original control point mesh. The points P 0,0, P 0,, P,0 and P,, which are shown in the figure below 6

7 !"# " are called face points, and are calculated as the aerage of the four points that bound the respectie face. If we define the face point F i,j to be the aerage of the points P i,j, P i+,j, P i,j+ and P i+,j+, then we can 7

8 rewrite the aboe equations with these face points substituted on the right-hand side, and obtain P 0,0 = F 0,0 P 0, = F 0,0 + F 0, + P 0, + P, P 0, = F 0, P 0,3 = F 0, + F 0, + P 0, + P, P,0 = F 0,0 + F,0 + P,0 + P, P, = F 0,0 + F 0, + F,0 + F, + P,0 + P 0, + 3P, + P, + P, 6 P, = F 0, + F, + P, + P, P,3 = F 0, + F 0, + F, + F, + P, + P 0, + 3P, + P, + P,3 6 P,0 = F,0 P, = F,0 + F, + P, + P, P, = F, P,3 = F, + F, + P, + P, P 3,0 = F,0 + F,0 + P,0 + P, P 3, = F,0 + F,0 + F, + F, + P,0 + P, + 3P, + P 3, + P, 6 P 3, = F, + F, + P, + P, P 3,3 = F, + F, + F, + F, + P, + P, + 3P, + P 3, + P,3 6

9 Simplifying these equations, we obtain P 0,0 = F 0,0 P 0, = F 0,0 + F 0, + P 0, + P, P 0, = F 0, P 0,3 = F 0, + F 0, + P 0, + P, P,0 = F 0,0 + F,0 + P,0 + P, P, = F 0,0 + F 0, + F,0 + F, + P,0 + P 0, + P, + P, + P, P, = F 0, + F, + P, + P, P,3 = F 0, + F 0, + F, + F, + P, + P 0, + P, + P, + P,3 P,0 = F,0 P, = F,0 + F, + P, + P, P, = F, P,3 = F, + F, + P, + P, P 3,0 = F,0 + F,0 + P,0 + P, P 3, = F,0 + F,0 + F, + F, + P,0 + P, + P, + P 3, + P, P 3, = F, + F, + P, + P, P 3,3 = F, + F, + F, + F, + P, + P, + P, + P 3, + P,3 In examining these equations, we see that the points P 0,, P 0,3, P,0, P,, P,, P,3, P 3,0 and P 3,, which are called edge points, are gien as the aerage of four points the two points that define the original edge and the two two new face points of the faces sharing the edge. This is shown in the following figure. 9

10 ! "#$% & '( ) * +,-./ : These edge points E i,j can be calculated either as E i,j = F i,j + F i,j + P i,j + P i+,j or E i,j = F i,j + F i,j + P i,j + P i,j+ depending on which side of the edge point the two faces lie. If we replace these edge points on the right-hand 0

11 side of the equations aboe, we obtain P 0,0 = F 0,0 P 0, = E 0, P 0, = F 0, P 0,3 = E 0, P,0 = E,0 P, = F 0,0 + F 0, + F,0 + F, + P,0 + P 0, + P, + P, + P, P, = E, P,3 = F 0, + F 0, + F, + F, + P, + P 0, + P, + P, + P,3 P,0 = F,0 P, = E, P, = F, P,3 = E, P 3,0 = E 3,0 P 3, = F,0 + F,0 + F, + F, + P,0 + P, + P, + P 3, + P, P 3, = E 3, P 3,3 = F, + F, + F, + F, + P, + P, + P, + P 3, + P,3 The remaining four points, P,, P,3, P 3, and P 3,3, as shown in the figure below!" #

12 are called ertex points. These points, as can be seen aboe, are somewhat complex, but after some reduction it can be seen that P i,j = Q + R + S where Q is the aerage of the face points of the faces adjacent to the ertex point, R is the aerage of the midpoints of the edges adjacent to the ertex point and S is the corresponding ertex from the original mesh. For example, if we consider the point P,3, then Q = F 0, + F 0, + F, + F, R = P 0, +P, + P,+P, + P,3+P, + P,+P, S = P, All sixteen points of the subdiision hae now been characterized in terms of face points, edge points and ertex points, and a geometric method has been deeloped to calculate these points. Extending this Subdiision Procedure to the Entire Patch We note that all 5 of the points can actually be calculated in this manner, as for example P, is a face point and can be calculated as the aerage of the four points bounding the face. In general, we call the mesh generated by the 5 points as a refinement of the original mesh. In this case, we can state the following rules to generate the points for the refinement of the surface: For each face in the original mesh, generate the new face points which are the aerage of all the original points defining the face. For each internal edge of the original mesh (i.e. an edge not on the boundary), generate the new edge points which are calculated as the aerage of four points: the two points which define the edge, and the two new face points for the faces that are adjacent to the edge. For each internal ertex of the original mesh (i.e. a ertex not on the boundary of the mesh), generate the new ertex points which are calculated as the aerage of Q, R and S, where Q is the aerage of the new face points of all faces adjacent to the original ertex point, R is the aerage of the midpoints of all original edges incident on the original ertex point, and S is the original ertex point.

13 The process generated from these rules actually extends to arbitrary rectangular meshes, so we can perform this process again on our refined mesh of 5 elements, producing a second refinement of the original mesh. In this case, we know that this represents yet another subdiision and that eentually, if we keep refining, this limit mesh will conerge to the original uniform B-spline surface. Thus, this process gies us a sequence of meshes, each of which is a refinement of the mesh directly aboe, and which conerges to the surface in the limit. For the purposes of rendering such a surface we can simply let the refinement process go until we hae a mesh that is sufficiently close to the actual surface and then utilize the mesh for rendering purposes. Summary The subdiision of the bicubic uniform B-spline surface produces a simple procedure based upon face points, edge points and ertex points, and can be extended to be a refinement procedure for an extended mesh based upon a rectangular topology. Catmull and Clark were able to take this procedure and produce a refinement strategy that works on a mesh of arbitrary topology. References ] CATMULL, E., AND CLARK, J. Recursiely generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design 0 (Sept. 97), All contents copyright (c) 996, 997, 99, 999, 000 Computer Science Department, Uniersity of California, Dais All rights resered. 3