Computer aided geometric design with Powell-Sabin splines

1. Computer aided geometric designwith Powell-Sabin splines Speaker: 周 联 2008.10.29 Ph.D Student Seminar

2. What is it? • C1-continuous • quadratic splines • defined on an arbitrary triangulation • in Bernstein-Bézier representation

3. Why use it? • PS-Splines vs. NURBS suited to represent strongly irregular objects • PS-Splines vs. Bézier triangles smoothness

4. Main works • M.J.D. Powell, M.A. Sabin. Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw., 3:316–325, 1977. • P. Dierckx, S.V. Leemput, and T. Vermeire. Algorithms for surface fitting using Powell-Sabin splines, IMA Journal of Numerical Analysis, 12, 271-299, 1992. • K. Willemans, P. Dierckx. Surface fitting using convex Powell-Sabin splines, JCAM, 56, 263-282,1994. • P. Dierckx. On calculating normalized Powell-Sabin B-splines. CAGD, 15(1):61–78, 1997. • J. Windmolders and P. Dierckx. From PS-splines to NURPS. Proc. of Curve and Surface Fitting, Saint-Malo, 45–54.1999. • E. Vanraes, J. Windmolders, A. Bultheel, and P. Dierckx. Automatic construction of control triangles for subdivided Powel-Sabin splines. CAGD, 21(7):671–682, 2004. • J. Maes, A. Bultheel. Modeling sphere-like manifolds with spherical Powell–Sabin B-splines. CAGD, 24 79–89, 2007. • H. Speleers, P. Dierckx, and S. Vandewalle. Weight control for modelling with NURPS surfaces. CAGD, 24(3):179–186, 2007. • D. Sbibih, A. Serghini, A. Tijini. Polar forms and quadratic spline quasi-interpolants on Powell–Sabin partitions. IMA Applied Numerical Mathematic, 2008. • H. Speleers, P. Dierckx, S. Vandewalle. Quasi-hierarchical Powell–Sabin B-splines. CAGD, 2008.

5. Authors Professor at Katholieke Universiteit Leuven(鲁汶大学), Computerwetenschappen. Paul Dierckx • Research Interests: • Splines functions, Powell-Sabinsplines. • Curves and Surface fitting. • Computer Aided Geometric Design. • Numerical Simulation.

6. Authors Stefan Vandewalle Professor at Katholieke Universiteit Leuven, Faculty of, CS • Research Projects: • Algebraic multigrid for electromagnetics. • High frequency oscillatory integrals and integral equations. • Stochastic and fuzzy finite element methods. • Optimization in Engineering. • Multilevel time integration methods.

7. Problem State (Powell,Sabain,1977) 9 conditions vs. 6 coefficients

8. A lemma

9. PS refinement Nine degrees of freedom

10. PS refinement The dimension equals 3n.

11. Other refinement

12. A theorem

13. Normalized PS-spline(Dierckx, 97) • Local support • Convex partition of unity. • Stability

14. Obtain the basis function Step 1.

15. Obtain the basis function Step 2.

16. Obtain the basis function Step 3.

17. Obtain the basis function Step 4.

18. PS-splines

19. Choice of PS triangles • To calculate triangles of minimal area • Simplify the treatment of boundary conditions

20. PS control triangles

21. PS control triangles

22. A Bernstein-Bézier representation

23. A Powell-Sabin surface

24. Local support(Dierckx,92)

25. Explicit expression for PS-splines

26. Normalized PS B-splines • Necessary and sufficient conditions:

27. The control points

28. The control points

30. The Bézier ordinates of a PS-spline

31. Spline subdivision(Vanraes, 2004) • Refinement rules of the triangulation

32. Refinement rules

35. Construction of refined control triangles

36. Triadically subdivided spline

37. Application • Visualization

41. QHPS(Speleers,08)

43. Data fitting

44. Data fitting

47. Rational Powell-Sabin surfaces