Non-Uniform Recursive Doo-Sabin Surfaces Zhangjin Huang'Guoping Wang University of Science and Technology of China Peking University,China SIAM Conference on Geometric and Physical Modeling ZHuang G Warg Nen-U ho
Non-Uniform Recursive Doo-Sabin Surfaces Zhangjin Huang1 Guoping Wang2 1University of Science and Technology of China 2Peking University, China SIAM Conference on Geometric and Physical Modeling Z Huang, G Wang Non-Uniform Recursive Doo-Sabin Surfaces (NURDSes)
Doo-Sabin Surfaces Generalization of uniform biquadratic B-spline surfaces to meshes of arbitrary topology [Doo and Sabin 1978]. Limit point rule:For an n-sided face, its centroid is the limit position of its associated extraordinary point. The extraordinary points are at the "centers"of n-sided faces. Convergence:The Doo-Sabin refinement is convergent for extraordinary points with arbitrary valence. Z Huang.G Wang Non-Uniform Recursive Doo-Sabin Surtaces (NURDSes)
Doo-Sabin Surfaces Generalization of uniform biquadratic B-spline surfaces to meshes of arbitrary topology [Doo and Sabin 1978]. Limit point rule: For an n-sided face, its centroid is the limit position of its associated extraordinary point. The extraordinary points are at the "centers" of n-sided faces. Convergence: The Doo-Sabin refinement is convergent for extraordinary points with arbitrary valence. Z Huang, G Wang Non-Uniform Recursive Doo-Sabin Surfaces (NURDSes)
Quadratic NURSSes Generalization of non-uniform biquadratic B-spline surfaces to meshes of arbitrary topology [Sederberg et al.1998]. No closed form limit point rules. Converge for n-sided faces with n<12,but may diverge if n 12 [Qin et al.1998]. Z Huang.G Wang Non-Uniform Recursive Doo-Sabin Surtaces (NURDSes)
Quadratic NURSSes Generalization of non-uniform biquadratic B-spline surfaces to meshes of arbitrary topology [Sederberg et al. 1998]. No closed form limit point rules. Converge for n-sided faces with n ≤ 12, but may diverge if n > 12 [Qin et al. 1998]. Z Huang, G Wang Non-Uniform Recursive Doo-Sabin Surfaces (NURDSes)
Doo-Sabin Subdivision n-1 p=>wP,i=0,n-1. j=0 Doo-Sabin version [Doo and Sabin 1978],extended to quadratic NURSS: i=j 3+2cos(2m-1/D,i≠j 4n Catmull-Clark variant [Catmull and Clark 1978]: 是+动，i-引=0 +,i-引=1 i-引>1 Z Huang.G Wang Non-Uniform Recursive Doo-Sabin Surfaces (NURDSes)
Doo-Sabin Subdivision Pi = Xn−1 j=0 wijPj , i = 0, . . . , n − 1. Doo-Sabin version [Doo and Sabin 1978], extended to quadratic NURSS: wij = ( n+5 4n , i = j 3+2 cos(2π(i−j)/n) 4n , i 6= j Catmull-Clark variant [Catmull and Clark 1978]: wij = 1 2 + 1 4n , |i − j| = 0 1 8 + 1 4n , |i − j| = 1 1 4n , |i − j| > 1 Z Huang, G Wang Non-Uniform Recursive Doo-Sabin Surfaces (NURDSes)
Catmull-Clark Variant of Doo-Sabin Subdivision Repeated averaging [Stam 2001,Zorin and Schroder 2001]: ■Linear subdivision: E,=B,+P+小 ■Dual averaging: E=R+E-1+B+) =（2+4 P++P+1+P)t∑ Z Huang.G Wang Non-Uniform Recursive Doo-Sabin Surfaces (NURDSes)
Catmull-Clark Variant of Doo-Sabin Subdivision Repeated averaging [Stam 2001, Zorin and Schröder 2001]: Linear subdivision: Ei = 1 2 (Pi + Pi+1), F = 1 n Xn−1 j=0 Pj Dual averaging: Pi = 1 4 (Pi + Ei−1 + Ei + F) = (1 2 + 1 4n )Pi + (1 8 + 1 4n )(Pi+1 + Pi−1) + 1 4n X |i−j|>1 Pj . Z Huang, G Wang Non-Uniform Recursive Doo-Sabin Surfaces (NURDSes)