Computer aided geometric design with Powell-Sabin splines Speaker : 周 联 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 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, P. Dierckx, S.V. Leemput, and T. Vermeire. Algorithms for surface fitting using Powell-Sabin splines, IMA Journal of Numerical Analysis, 12, , K. Willemans, P. Dierckx. Surface fitting using convex Powell-Sabin splines, JCAM, 56, ,1994. P. Dierckx. On calculating normalized Powell-Sabin B-splines. CAGD, 15(1):61–78, J. Windmolders and P. Dierckx. From PS-splines to NURPS. Proc. of Curve and Surface Fitting, Saint- Malo, 45– E. Vanraes, J. Windmolders, A. Bultheel, and P. Dierckx. Automatic construction of control triangles for subdivided Powel-Sabin splines. CAGD, 21(7):671–682, J. Maes, A. Bultheel. Modeling sphere-like manifolds with spherical Powell–Sabin B-splines. CAGD, 24 79–89, H. Speleers, P. Dierckx, and S. Vandewalle. Weight control for modelling with NURPS surfaces. CAGD, 24(3):179–186, D. Sbibih, A. Serghini, A. Tijini. Polar forms and quadratic spline quasi-interpolants on Powell–Sabin partitions. IMA Applied Numerical Mathematic, 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

A lemma

PS refinement Nine degrees of freedom

PS refinement The dimension equals 3n.

Other refinement

A theorem

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.

PS-splines

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

PS control triangles

A Bernstein-Bézier representation

A Powell-Sabin surface

Local support(Dierckx,92)

Explicit expression for PS-splines

Normalized PS B-splines Necessary and sufficient conditions:

The control points

The Bézier ordinates of a PS-spline

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

Refinement rules

Construction of refined control triangles

Triadically subdivided spline

Application Visualization

QHPS(Speleers,08)

Data fitting

Rational Powell-Sabin surfaces

B-spline representation for PS splines on the sphere(Maes,07)

Thank you!