1. Normal based subdivision scheme for curve and surface design 杨勋年 2004.12 http://www.math.zju.edu.cn/yxn

2. What is CAGD Computer science CAGD Engineering mathematics

3. Content What is subdivision? - corner cutting algorithms - interpolating subdivision Normal based subd. Scheme - the scheme - for curve design - for surface design Summary

4. What is subdivision Recursive refinement for the generation of - functions (approx. theory, wavelet) - curves and surfaces (CAGD) Classification - Steady vs nonsteady - rational vs nonrational - Linear vs nonlinear

5. Corner cutting algorithms Corner cutting: Chaikin, B-spline Convergence: de Boor, Riesenfeld, Gregory, et al

6. Subdivision of B-spline Uniform cubic B-spline Derive the rule by knots insertion

7. Arbitrary control mesh The topological rule The geometric rule Catmull-clark scheme

8. Catmull-clark subdivision surface

9. Interpolating subdivision Edge split Vertex refinement

10. Four-point scheme Cubic precision (Dyn, et al 1987) Linear subdivision Add a point by local cubic curve interpolation A geometric look at four point scheme

11. Butterfly scheme Extension of 4-point scheme (Dyn, et al 1990) Triangular control mesh (1 to 4) Local bicubic surface interpolation Control mesh Parametric domain

12. Limitations Interpolating or fitting - efficient representation - scanning data processing By CC scheme - solve inverse problem By butterfly scheme - not fair - not easy for normal control

13. Content What is subdivision? - corner cutting algorithms - interpolating subdivision Normal based subd. Scheme - the scheme - for curve design - for surface design Summary

14. Our approach Normal refinement - for each vertex for each level Vertex refinement - subdivide each edge - project sub-edges onto normals - compute displacement vector - compute new vertex

15. The basic scheme

16. Normal refinement Fixed normal at selected vertexes - the normal will be interpolated Refine other normal for each subdivision The rule for normal computation - chord tangent angles are close

17. Normal computation Curve caseSurface case

18. Convergence Active chord tangent angles - converge to zero - within fixed scale Fixed chord tangent angles - are bounded - convergence Polygon series - converge - tangent continuous

19. For curve design The freedoms - subd. ratio of edges - scale for displacement vector Shape preserving - same scheme - explicit choices of freedoms

20. Shape preserving scheme

21. Freeform curve

22. Bottle design Control polygon Subdivision curve

23. For surface design Triangular control mesh Topology split Vertex refinement - Normal based scheme

24. Topology split

25. Head model Control mesh Subdivision surface

26. Solid star Control meshSubdivision surface Butterfly subdivision surfaceModified butterfly subd. surface

27. Knot surface Control mesh Butterfly subd.Normal based subd.

28. Summary Normal based subdivision - a geometric scheme - tangent continuous - natural shape Contributions - normal refinement as well as vertex refinement - geometric dependent instead of parametric dependent

29. Thank you !