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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 designwith Powell-Sabin splines Speaker: 周 联 2008.10.29 Ph.D Student Seminar
What is it? • C1-continuous • quadratic splines • defined on an arbitrary triangulation • in Bernstein-Bézier representation
Why use it? • PS-Splines vs. NURBS suited to represent strongly irregular objects • PS-Splines vs. Bézier triangles smoothness
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.
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.
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.
Problem State (Powell,Sabain,1977) 9 conditions vs. 6 coefficients
PS refinement Nine degrees of freedom
PS refinement The dimension equals 3n.
Normalized PS-spline(Dierckx, 97) • Local support • Convex partition of unity. • Stability
Obtain the basis function Step 1.
Obtain the basis function Step 2.
Obtain the basis function Step 3.
Obtain the basis function Step 4.
Choice of PS triangles • To calculate triangles of minimal area • Simplify the treatment of boundary conditions
Normalized PS B-splines • Necessary and sufficient conditions:
Spline subdivision(Vanraes, 2004) • Refinement rules of the triangulation
Application • Visualization