Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Graphics Programming Lee Byung-Gook Dongseo Univ.

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Presentation transcript:

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Graphics Programming Lee Byung-Gook Dongseo Univ.

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Affine combination Linear combinations Affine(Barycentric) combinations Convex combinations Barycentric coordinates

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Affine combination Euclidean coordinate system Coordinate-free system

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Polynomial interpolation

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Polynomial interpolation Lagrange polynomials

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Examples of cubic interpolation

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Bezier

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Representation Bezier

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., 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., 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., Piecewise cubic hermite interpolation

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Cubic spline interpolation

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Cubic spline interpolation

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Spline interpolation based on the 1-norm Cubic Spline Interpolation with Natural boundary condition

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., 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., 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., Linear splines

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Quadratic splines

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Quadratic splines

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Representation splines

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., B-spline Recurrence Relation Bernstein polynomial

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., B-spline

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., B-spline Smoothness=Degree-Multiplicity

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., B-spline basis functions

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Refinement relation for B-spline

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Repeated integration for B-spline

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Truncated powers for B-spline

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Cross-sectional Volumes

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Cross-sectional Volumes for subcubes

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Box-spline as Cross-sectional Volumes

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Bivariate Box spline over triangular grid

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Spline space

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Univariate spline

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Cubic splines

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Condition number

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., 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)

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Stability

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Blossom

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., Blossom

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., 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)