Spline Methods in CAGD

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Spline Methods in CAGD

• byung-gook lee
• Dongseo Univ.
• lbg_at_dongseo.ac.kr
• http//kowon.dongseo.ac.kr/lbg/

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Contents
http//www.subdivision.org/subdivision/index.jsp
http//www.uio.no/studier/emner/matnat/ifi/INF-MAT
5340/v04/litteratur.xml
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http//www.cgal.org/

• Affine combination
• Lagrange
• Bezier
• Spline
• B-spline
• E-spline
• Box-spline
• Refinement relation
• Cross-sectional volumes
• Subdivision schemes

Reference Spline Methods Draft Tom Lyche and
Knut Morken Subdivision Methods for Geometric
Design A Constructive Approach Joe Warren and
Henrik Weimer
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Affine combination

• Linear combinations
• Affine(Barycentric) combinations
• Convex combinations
• Barycentric coordinates

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Affine combination

• Euclidean coordinate system

Coordinate-free system
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Polynomial interpolation
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Polynomial interpolation

• Lagrange polynomials

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Examples of cubic interpolation
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Bezier
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Representation Bezier
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Bezier

• Paul de Faget de Casteljau, Citroen, 1959
• Pierre Bezier, Renault, UNISUF system, 1962
• A.R. Forrest, Cambridge, 1968

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Properties of Bezier

• Affine invariance
• Convex hull property
• Endpoint interpolation
• Symmetry
• Linear precision
• Pseudo-local control
• Variation Diminishing Property

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Spline curves

• J. Ferguson , Boeing Co., 1963
• C. de Boor, W. Gordon, General Motors, 1963
• to interpolate given data
• piecewise polynomial curves with certain
  differentiability constraints
• not to design free form curves

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Piecewise cubic hermite interpolation
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Cubic spline interpolation
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Linear B-spline
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Quadratic B-spline
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Quadratic B-spline
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B-spline

• Recurrence Relation

• Bernstein polynomial

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B-spline
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B-spline

• SmoothnessDegree-Multiplicity

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Representation B-spline
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Cubic B-spline
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B-spline space
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B-spline

• C. de Boor, 1972
• W. Gordon, Richard F. Riesenfeld, 1974
• Larry L. Schumaker
• Tom Lyche
• Nira Dyn

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B-spline basis functions
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Refinement relation for B-spline
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Repeated integration for B-spline
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E-splines
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E-splines
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Truncated powers for B-spline
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Cross-sectional Volumes
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B-spline basis as Cross-sectional Volumes
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Cross-sectional Volumes for subcubes
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Bivariate Tensor Product B-spline
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Box-spline as Cross-sectional Volumes
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Bivariate Box spline over triangular grid
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Shadows of boxes
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Refinement Relation for B-spline
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Refinement Relation for Box-spline
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Refinement Relation for Box-spline
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Subdivision for Box-spline
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Refinement Relation for Box-spline
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Subdivision for Box-spline
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Refinement Relation for Box-spline
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Refinement Relation for Box-spline
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Refinement Relation for Box-spline
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Condition number
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Condition number of B-spline basis

• Tom Lyche and Karl Scherer, On the p-norm
  condition number of the multivariate triangular
  Bernstein basis, Journal of Computational and
  Applied Mathematics 119(2000) 259-273

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Stability
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Blossom
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Blossom
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B-spline problems

• Degree Elevation
• Degree Reduction
• Knot Insertion
• Knot Deletion

Gerald Farin, Curves and Surfaces for Computer
Aided Geometric Design, 4th ed, Academic Press
(1996) Ronald N. Goldman, Tom Lyche, editors,
Knot Insertion and Deletion Algorithms for
B-Spline Curves and Surfaces, SIAM (1993)
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Bezier Degree Reduction
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Bezier Degree Reduction

• Least square method
• Legendre-Bernstein basis transformations

Rida T. Farouki, Legendre-Bernstein basis
transformations, Journal of Computational and
Applied Mathematics 119(2000) 145-160. Byung-Gook
Lee, Yunbeom Park and Jaechil Yoo, Application of
Legendre-Bernstein basis transformations to
degree elevation and degree reduction, Computer
Aided Geometric Design 19(2002) 709-718.
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Bezier Degree Reduction with constrained
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Contents

• Affine combination
• Bezier curves
• Spline curves
• B-spline curves
• Condition number
• L1-norm spline
• Quasi-interpolant
• Box-spline
• Cross-sectional volumes
• Refinement relation
• Subdivision

Reference Spline Methods Draft Tom Lyche and
Knut Morken Subdivision Methods for Geometric
Design A Constructive Approach Joe Warren and
Henrik Weimer