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. Bsplines
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 and Calculus (suggested).
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.