Distance Between a Catmull- Clark Subdivision Surface and Its Limit Mesh Zhangjin Huang,Guoping Wang Peking University,China
Distance Between a CatmullClark Subdivision Surface and Its Limit Mesh Zhangjin Huang, Guoping Wang Peking University, China
Catmull-Clark subdivision surface (CCSS Generalization of uniform bicubic B-spline surface C2 continuous except at extraordinary points The limit of a sequence of recursively refined control meshes initial mesh step 1 limit surface
Generalization of uniform bicubic B-spline surface ◼ continuous except at extraordinary points The limit of a sequence of recursively refined control meshes Catmull-Clark subdivision surface (CCSS) initial mesh step 1 limit surface 2 C
CCSS patch:regular vs.extraordinary Blue:regular Red:extraordinary Control mesh Limit surface Assume each mesh face in the control mesh ■a quadrilateral at most one extraordinary point (valence n is not 4) An interior mesh face F in the control mesh-a surface patch S in the limit surface Regular patch:bicubic B-spline patches,16 control points Extraordinary patch:not B-spline patches,2n+8 control points
CCSS patch: regular vs. extraordinary Assume each mesh face in the control mesh ◼ a quadrilateral ◼ at most one extraordinary point (valence n is not 4) An interior mesh face in the control mesh → a surface patch in the limit surface ◼ Regular patch: bicubic B-spline patches, 16 control points ◼ Extraordinary patch: not B-spline patches, 2n+8 control points Control mesh Limit surface S F Blue: regular Red: extraordinary
Control mesh approximation and error 无法显示该图片 Control mesh is a piecewise linear approximation to a CCSS Approximation error:the maximal distance between a CCSS and the control mesh Distance between a CCSS patch S and its mesh face F(or control mesh)is defined as ()-Fu is unit square =[0,1]x[0,1] S(u,v)is Stam's parametrization of S over F(u,v)is bilinear parametrization of F over
Control mesh approximation and error Control mesh is a piecewise linear approximation to a CCSS Approximation error: the maximal distance between a CCSS and the control mesh Distance between a CCSS patch and its mesh face (or control mesh) is defined as ◼ is unit square ◼ is Stam’s parametrization of over ◼ is bilinear parametrization of over S( , ) u v S F(,) u v S F F ( , ) max ( , ) ( , ) u v u v u v S F− = [0,1] [0,1]
Distance bound for control mesh approximation The distance between a CCSS patch S and its control mesh is bounded as Cheng et al.2006] maxS(u,)-F(w,v)‖≤C(n)M (u,v)EQ C(n)is a constant that only depends on valence n M is thethe second order norm of 2n+8 control points of S 2n+1 2m+8 ■For regular patches, C(4)=1 3 n+6 2n+3 2n+2
Distance bound for control mesh approximation The distance between a CCSS patch and its control mesh is bounded as [Cheng et al. 2006] ◼ is a constant that only depends on valence n ◼ is the the second order norm of 2n+8 control points of ◼ For regular patches, S C n( ) M S (4) 1 3 C = ( , ) max ( , ) ( , ) ( ) u v u v u v C n M S F−