Graphics Programming

Graphics Programming 2003, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Lee Byung-GookDongseo 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

Polynomial interpolation

• Lagrange polynomials

Examples of cubic interpolation

Bezier

Representation Bezier

Properties of Bezier

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

Bezier

• Paul de Faget de Casteljau, Citroen, 1959• Pierre Bezier, Renault, UNISUF system, 1962• A.R. Forrest, Cambridge, 1970

Piecewise cubic hermite interpolation

Cubic spline interpolation

Spline interpolation based on the 1-norm

Cubic Spline Interpolation with Natural boundary condition

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

B-spline

• C. de Boor, 1972• W. Gordon, Richard F. Riesenfeld, 1974

• Larry L. Schumaker• Tom Lyche• Nira Dyn

Linear splines

Quadratic splines

Representation splines

B-spline

• Recurrence Relation

• Bernstein polynomial

B-spline

• Smoothness=Degree-Multiplicity

B-spline basis functions

Refinement relation for B-spline

Repeated integration for B-spline

Truncated powers for B-spline

Cross-sectional Volumes

Cross-sectional Volumes for subcubes

Box-spline as Cross-sectional Volumes

Bivariate Box spline over triangular grid

Spline space

Univariate spline

Cubic splines

Condition number

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

Stability

Blossom

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)

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