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
The course provide to the students the basic notions for the theoretical understanding of numerical methods for computer aided geometric design with focus on parametric curves and surfaces in Bézier and B-spline form and related algorithms for geometric modelling and numerical approximations.
At the end of the course, the studente will be able to:
- understand and present the mathematical formulation of the proposed problems and the relation with the corresponding numerical solution;
- understand and present the mathematical aspects guaranteeing the efficiency and accuracy of the numerical methods;
- solve some test problems by writing in Matlab programs implementing the studied methods.
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
At the end of the course a topic for the development of a project which includes the implementation of geometric modelling algorithms and their numerical testing. The final oral test consists of a series of questions to verify the knowledge of the theoretical mathematical aspects of the numerical methods. The evaluation, which takes into account the results of the project and of the final oral test, is focused on assessing the knowledge of the mathematical concepts the methods are based on and of the ability to use them to solve different application problems.
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.