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Computer aided geometric design with Powell-Sabin splines. Speaker : 周 联 2008.10.29. Ph.D Student Seminar. What is it?. C 1 -continuous quadratic splines defined on an arbitrary triangulation in Bernstein-Bézier representation. Why use it?. PS-Splines vs. NURBS

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Computer aided geometric design with Powell-Sabin splines


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    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

    29. The Bézier ordinates of a PS-spline

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

    31. Refinement rules

    32. Construction of refined control triangles

    33. Triadically subdivided spline

    34. Application • Visualization

    35. QHPS(Speleers,08)

    36. Data fitting

    37. Data fitting

    38. Rational Powell-Sabin surfaces