Computer aided geometric design with PowellSabin splines

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Computer aided geometric designwith
Powell-Sabin splines

• Speaker ? ?
• 2008.10.29

Ph.D Student Seminar
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What is it?

• C1-continuous
• quadratic splines
• defined on an arbitrary triangulation
• in Bernstein-Bézier representation

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Why use it?

• PS-Splines vs. NURBS
• suited to represent strongly irregular objects
• PS-Splines vs. Bézier triangles
• smoothness

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Main works

• M.J.D. Powell, M.A. Sabin. Piecewise quadratic
  approximations on triangles. ACM Trans. Math.
  Softw., 3316325, 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)6178, 1997.
• J. Windmolders and P. Dierckx. From PS-splines to
  NURPS. Proc. of Curve and Surface Fitting,
  Saint-Malo, 4554. 1999.
• E. Vanraes, J. Windmolders, A. Bultheel, and P.
  Dierckx. Automatic construction of control
  triangles for subdivided Powel-Sabin splines.
  CAGD, 21(7)671682, 2004.
• J. Maes, A. Bultheel. Modeling sphere-like
  manifolds with spherical PowellSabin B-splines.
  CAGD, 24 7989, 2007.
• H. Speleers, P. Dierckx, and S. Vandewalle.
  Weight control for modelling with NURPS surfaces.
  CAGD, 24(3)179186, 2007.
• D. Sbibih, A. Serghini, A. Tijini. Polar forms
  and quadratic spline quasi-interpolants on
  PowellSabin partitions. IMA Applied Numerical
  Mathematic, 2008.
• H. Speleers, P. Dierckx, S. Vandewalle.
  Quasi-hierarchical PowellSabin B-splines. CAGD,
  2008.

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Authors
Paul Dierckx

• Professor at
• Katholieke Universiteit Leuven(????),
• Computerwetenschappen.

• Research Interests
• Splines functions, Powell-Sabinsplines.
• Curves and Surface fitting.
• Computer Aided Geometric Design.
• Numerical Simulation.

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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.

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Problem State (Powell,Sabain,1977)
9 conditions vs. 6 coefficients
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A lemma
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PS refinement
Nine degrees of freedom
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PS refinement
The dimension equals 3n.
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Other refinement
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A theorem
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Normalized PS-spline(Dierckx, 97)

• Local support
• Convex partition of unity.
• Stability

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Obtain the basis function
Step 1.
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Obtain the basis function
Step 2.
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Obtain the basis function
Step 3.
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Obtain the basis function
Step 4.
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PS-splines
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Choice of PS triangles

• To calculate triangles of minimal area
• Simplify the treatment of boundary conditions

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PS control triangles
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PS control triangles
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A Bernstein-Bézier representation
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A Powell-Sabin surface
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Local support(Dierckx,92)
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Explicit expression for PS-splines
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Normalized PS B-splines

• Necessary and sufficient conditions

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The control points
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The control points
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The Bézier ordinates of a PS-spline
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Spline subdivision(Vanraes, 2004)

• Refinement rules of the triangulation

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Refinement rules
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Construction of refined control triangles
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Triadically subdivided spline
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Application

• Visualization

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QHPS(Speleers,08)
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Data fitting
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Data fitting
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Rational Powell-Sabin surfaces
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B-spline representation for PS splines on the
sphere(Maes,07)
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• Thank you!