Polynomial and rational Bézier curves. Bézier-spline curves. B-spline curves and NURBS. Bézier and B-spline tensor-product surfaces and rational extensions. Barycentric coordinates. Triangular Bézier patches. Geometric continuity. Algorithms for the design of free-form curves and surfaces.
- C de Boor (2001), A Practical Guide to Splines, Revised edition, Applied Mathematical Sciences 27, Springer-Verlag, New York.
- G. Farin (2002), Curves and Surfaces for CAGD: a practical guide, Kaufmann series in computer graphics and geometric modelling, San Francisco M. Kaufmann Publishers.
Learning Objectives
Knowledge: basic methods of Computer-Aided-Geometric-Design with particular focus on splines.
Capabilities: development and usage of algorithms for computer design of free-form curves and surfaces.
Prerequisites
Basic elements of Numerical calculus (finite arithmetic, numerical methods for linear systems). Knowledge of Matlab.
Teaching Methods
48 hours of teaching. About 1/4 of them consists in computer implementations and experiments.
Type of Assessment
Oral exam and discussion of a student work developed in Matlab.
Course program
Hints of differential geometry. Plynomial and rational Bézier curves and related algorithms (De casteljau, subdivision, degree elevation). Bézier-splines and their application to Hermite interpolation. Geometric continuity.
C^0,C^1,C^2,G^1,G^2 continuity. The B-spline basis with simple and multiple knots. B-spline curves and NURBS extension. De Boor, degree elevation and knot insertion algorithms. Basic subdivision algoritms. Bezier and B-spline tensor-product patches, C^0, C^1 and G^1 continuity between two patches. Bicubic Hermite interpolation. Barycentric coordinates. Triangular Bézier patches and related algorithms. The Powell-Sabin macroelement for interpolation.