1. Distance Between a Catmull- Clark Subdivision Surface and Its Limit Mesh Zhangjin Huang, Guoping Wang Peking University, China

2.  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 1limit surface

3. 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 meshLimit surface Blue: regular Red: extraordinary

4. 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

5. 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,

6. Limit mesh approximation  An interior mesh face → a limit face → a surface patch  We derive a bound on the distance between a patch and its limit face (or limit mesh) as means that the limit mesh approximates a CCSS better than the control mesh  Limit mesh : push the control points to their limit positions.  It inscribes the limit surface

7. Regular patches: how to estimate  Regular patch is a bicubic B-spline patch:  Limit face, then  It is not easy to estimate directly!

8. Transformation into bicubic Bézier forms  Both and can be transformed into bicubic Bézier form :

9. Regular patches: distance bound  Core idea: Measure through measuring

10. Regular patches: distance bound (cont.)  Bound with the second order norm, it follows that  Distance bound function of with respect to is Diagonal  By symmetry,

11. Regular patch: distance bound (cont.)  Theorem 1 The distance between a regular CCSS patch and the corresponding limit face is bounded by The distance between a regular patch and its corresponding mesh face is bounded as [Cheng et al 2006]

12. Extraordinary patches: parametrization  An extraordinary patch can be partitioned into an infinite sequence of uniform bicubic B-spline patches  Partition the unit square into tiles  Stam ’ s parametrization: Transformation maps the tile onto the unit square

13. Extraordinary patches: distance bound  Limit face can be partitioned into bilinear subfaces defined over :  Similar to the regular case, for  By solving 16 constrained minimization problems, we have

14. Extraordinary patch: distance bound function  Thus  is the distance bound function of with respect to :  The distance bound function of with respect to is defined as: Diagonal  By symmetry,

15. Extraordinary patches: distance bound  Theorem 2 The distance between an extra- ordinary CCSS patch and the corresponding limit face is bounded by  has the following properties:, attains its maximum in  Only needed to consider 2 subpatches and

16. Extraordinary patches: bound constant , , strictly decreases as n increases

17. Comparison of bound constants  First two lines are for control mesh approx.  Last line are for limit mesh approximation , n345678910 Cheng et al. 06 0.7848140.3333330.5748900.6422670.5273570.5824360.5101810.678442 Huang et al. 06 0.7843140.3333330.5748900.5490200.5273570.4242420.5101810.519591 Limit mesh 0.2581460.250.2431290.2374540.2327610.2288480.2255490.222738

18. Application to adaptive subdivison Error tolerance Frog modelCar model Control meshLimit meshControl meshLimit mesh 0.125,64214,52714,7178,285 0.0543,86426,48224,90514,375 0.01172,159113,71397,70458,805 The number of faces decreases by about 30%

19. Application to CCSS intersection

20. Conclusion  Propose an approach to derive a bound on the distance between a CCSS and its limit mesh Our approach can be applied to other spline based subdivision surfaces  Show that a limit mesh may approximate a CCSS better than the corresponding control mesh

21. Thank you!