Computer aided geometric designwith Powell-Sabin splinesSpeaker: 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-Ma

lo, 45–54. 1999. E. Vanraes, J. Windmolders, A. Bultheel, and P. Dierckx. Automatic construction of control triangles for s

ubdivided 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 7

9–89, 2007. H. Speleers, P. Dierckx, and S. Vandewalle. Weight control for modelling with NURPS surfaces. CAGD, 2

4(3):179–186, 2007. D. Sbibih, A. Serghini, A. Tijini. Polar forms and quadratic spline quasi-interpolants on Powell–Sabin parti

tions. IMA Applied Numerical Mathematic, 2008. H. Speleers, P. Dierckx, S. Vandewalle. Quasi-hierarchical Powell–Sabin B-splines. CAGD, 2008.

Authors

Professor atKatholieke 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 coefficients22),( fveuvducvbuavu

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 functionStep 2.

Obtain the basis functionStep 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

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 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

Data fitting

Rational Powell-Sabin surfaces

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

Thank you!