Polynomial and rational Bézier curves. Algorithms: de Casteljau, degree elevation, and subdivision. Rational Bézier form of conic sections. Spline functions. Bézier spline curves. Classic and geometric continuity. B-splines with single and multiple knots. B-spline curves. Algorithms: de Boor and knot insertion. Tensor-product and triangular Bézier patches. Tensor-product B-spline surfaces.
J. Hoschek and D. Lasser, "Fundamentals of Computer Aided Geometric Design", translated from the original German edition by L.L. Schumaker, A.K. Peters, Wellesley, MA, 1993.
G. Farin, "Curves and Surfaces for Computer Aided Geometric Design". A practical guide, Academic Press, Boston, MA, 1993.
Learning Objectives
Spline functions and their compact representation in B-form. Curve fitting Matlab Toolbox. Parametric representations of curves and surfaces. Numerical methods for computer aided geometric design. Geometric continuity.
Students will be able to develop algorithms for curve and surface shape design, by also exploiting the functionalities of Matlab functions dedicated to splines.
Prerequisites
Basics of Numerical Analysis.
Teaching Methods
Classroom lectures and programming exercises. The exercises are dedicated to the developement and testing of the methods covered in the classroom.
Type of Assessment
Oral examination with open questions and exercises on the topics of the program. This also includes the discussion of Matlab code and related numerical examples.
Course program
Basics of differential geometry of curves and surfaces. Polynomial and rational Bézier curves. Algorithms: de Casteljau, degree elevation, and subdivision. Rational Bézier form of conic sections. Classic and geometric continuity. B-splines on single and multiple knots. B-spline curves. Algorithms: de Boor and knot insertion. Tensor-product and triangular Bézier patches. Tensor-product B-spline surfaces. Hierarchical B-splines.