Graphics Programming
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Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected] Graphics Programming Lee Byung-Gook Dongseo Univ. http://kowon.dongseo.ac.kr/~lbg/
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Graphics Programming. Lee Byung-Gook Dongseo Univ. http://kowon.dongseo.ac.kr/~lbg/. Affine combination. Linear combinations Affine(Barycentric) combinations Convex combinations Barycentric coordinates. Affine combination. Euclidean coordinate system. Coordinate-free system. - PowerPoint PPT Presentation
Transcript of Graphics Programming
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Graphics Programming
Lee Byung-GookDongseo Univ.
http://kowon.dongseo.ac.kr/~lbg/
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Affine combination
• Linear combinations
• Affine(Barycentric) combinations
• Convex combinations
• Barycentric coordinates
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Affine combination
Euclidean coordinate system
Coordinate-free system
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Polynomial interpolation
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Polynomial interpolation
• Lagrange polynomials
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Examples of cubic interpolation
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Representation Bezier
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Properties of Bezier
• Affine invariance• Convex hull property• Endpoint interpolation• Symmetry• Linear precision• Pseudo-local control• Variation Diminishing Property
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Bezier
• Paul de Faget de Casteljau, Citroen, 1959• Pierre Bezier, Renault, UNISUF system, 1962• A.R. Forrest, Cambridge, 1970
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Piecewise cubic hermite interpolation
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Cubic spline interpolation
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Cubic spline interpolation
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Spline interpolation based on the 1-norm
Cubic Spline Interpolation with Natural boundary condition
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
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
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
B-spline
• C. de Boor, 1972• W. Gordon, Richard F. Riesenfeld, 1974
• Larry L. Schumaker• Tom Lyche• Nira Dyn
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Quadratic splines
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Quadratic splines
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Representation splines
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
B-spline
• Recurrence Relation
• Bernstein polynomial
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
B-spline
• Smoothness=Degree-Multiplicity
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
B-spline basis functions
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Refinement relation for B-spline
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Repeated integration for B-spline
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Truncated powers for B-spline
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Cross-sectional Volumes
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Cross-sectional Volumes for subcubes
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Box-spline as Cross-sectional Volumes
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Bivariate Box spline over triangular grid
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
Univariate spline
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
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
Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:[email protected]
B-spline problems
• Degree Elevation• Degree Reduction• Knot Insertion• Knot Deletion
Gerald Farin, Curves and Surfaces for Computer Aided Geometric Design, 4 th ed, Academic Press (1996)Ronald N. Goldman, Tom Lyche, editors, Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces, SIAM (1993)
