BIQUADRATIC UNIFORM B-SPLINE SURFACE REFINEMENT

Transcription

1 On-Line Geometric Modeling Notes BIQUADRATIC 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. In these notes we present a study of the quadratic uniform B-spline case, and deelop the refinement rules that can be generalized into the Doo-Sabin subdiision method. The Matrix Equation for a Uniform Biquadratic Spline Surface Consider the biquadratic uniform B-spline surface P(u, ) defined by the 3 3 array of control points P P, P, P, P, P, P, P, P, P,

2 ! " #$ % & and the following equation (in matrix form) P(u, ) u u MP M T where M is the 3 3 matrix M The matrix M defines the quadratic uniform B-spline blending functions when multiplied by. Subdiiding the Surface This patch can be subdiided into four subpatches, which can be generated from 6 unique sub-control points. We focus on the subdiision scheme for only one of the four (the subpatch corresponding to u, ), as the others will follow by symmetry. The following figure illustrates the 6 points produced by subdiiding into four subpatches. We hae outlined the initial subpatch that we consider below. It should be noted that the four interior control points are utilized by each of the four subpatch components of the subdiision.

3 To subdiide the surface, we consider the reparameterization of the surface by u u and and 3

4 define this new surface as P (u, ). Substituting these into the equation, we obtain P (u, ) P( u, ) u ( u ) MP M T ( ) u u MP M T T u u MM MP M T (M ) T M T T u u M M M P M T (M ) T M T u u M M M P M M u u MP M T T T 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 MSM T for some 3 3 control point array S. This implies that P (u, ) is a uniform biquadratic B-spline patch. The matrix S is typically called the splitting matrix, and is gien by S and so the control point mesh P corresponding to the subdiided patch is related to the original control points mesh by P SP S T By carrying out the algebra, we hae that T P 3 P, P, P, 3 3 P, P, P, 3 3 P, P, P, 3 3P, + P, 3P, + P, 3P, + P, 3 6 P, + 3P, P, + 3P, P, + 3P, 3 3 3P, + P, 3P, + P, 3P, + P, P, P, P, 6 P, P, P, P, P, P, 5

6 where the P i,j can be written as P, 6 (3(3P, + P, ) + (3P, + P, )) P, 6 ((3P, + P, ) + 3(3P, + P, )) P, 6 (3(3P, + P, ) + (3P, + P, )) P, 6 (3(P, + 3P, ) + (P, + 3P, )) P, 6 ((P, + 3P, ) + 3(P, + 3P, )) P, 6 (3(P, + 3P, ) + (P, + 3P, )) P, 6 (3(3P, + P, ) + (3P, + P, )) P, 6 ((3P, + P, ) + 3(3P, + P, )) P, 6 (3(3P, + P, ) + (3P, + P, )) These equations can be looked at in two ways:. Each of these points P i,j utilizes the four points on a certain face of the rectangular mesh, and calculates a new point by weighing the four points in the ratio of Thus, this algorithm can be specified by using subdiision masks, which specify the ratios of the points on a face to generate the new points. In this case, the subdiision masks are as follows. Each of these equations is built from weighing the points on an edge in the ratio of 3-. and then weighing the resulting points in the ratio 3-. These are exactly the ratios of Chaikin s cure and so this method can be looked upon as a extension of Chaikin s cure to surfaces. Generating the Refinement Procedure 6

7 To generate the subdiision surface, we consider all 6 of the possible points generated through the binary subdiision of the quadratic patch. It is easily seen that each of these points can be generated through considering other subdiisions of the patch P (u, ) and can be defined by the same subdiision masks Summary The extension of Chaikin s Cure to surfaces is quite straightforward. The analysis has produced a simple mask that can be utilized to define the new points on each face (one per ertex). Connecting up these ertices into a new mesh is straightforward. The interesting extension of this algorithm is when the control point mesh does not hae a rectangular topological structure. In this case, we can still utilize the same paradigm and this was accomplished by Donald Doo and Malcolm Sabin in their Doo-Sabin surfaces References CHAIKIN, G. An algorithm for high speed cure generation. Computer Graphics and Image Processing 3 (97), DOO, D. A subdiision algorithm for smoothing down irregularly shaped polyhedrons. In Proced. Int l Conf. Ineractie Techniques in Computer Aided Design (978), pp Bologna, Italy, IEEE Computer Soc. All contents copyright (c) 996, 997, 998, 999, Computer Science Department, Uniersity of California, Dais All rights resered. 7