Mathematical Methods for CAD

Description

This book sets out the fundamental elements of the theory of computational geometry and computer-aided design in a mathematically rigorous manner. Splines and Bézier curves are first tackled, leading to Bézier surfaces, triangulation, and box splines. The final chapter is devoted to algebraic geometry and provides a firm theoretical basis for anyone wishing to seriously develop and investigate CAD systems Content: Ch. 1. B-splines. 1.1. Piecewise polynomial functions. 1.2. An example: Cubic splines. 1.3. B-splines: Fundamental properties. 1.4. The B-spline basis of P[subscript k, [tau], r]. 1.5. Basic Algorithms for B-splines. 1.6. Approximation by a spline curve. 1.7. Divided differences -- Ch. 2. Spline curves and Bezier curves. 2.1. Bernstein polynomials. 2.2. B-spline curves. 2.3. Algorithms for spline curves -- Ch. 3. Interpolation and complements. 3.1. Interpolation. 3.2. Other properties of the matrix N. 3.3. Matrix representation. 3.4. Junction between two curves. 3.5. Rational curves -- Ch. 4. Spline Surfaces. 4.1. Tensor products. 4.2. Particular case of Bezier surfaces. 4.3. Interpolation and approximation. 4.4. Bernstein polynomials. 4.5. Triangular Bezier patches. 4.6. Junction between Bezier patches. 4.7. Base points of rational Bezier patches. 4.8. Polyhedral splines. 4.9. Box splines -- Ch. 5. Triangulations. 5.1. Voroni diagram. 5.2. Generalities about triangulations. 5.3. Delaunay triangulations. 5.4. Construction Algorithm. 5.5. Remarks about complexity -- Ch. 6. Notions of real algebraic geometry. 6.1. Roots of one variable polynomials. 6.2. Resultants and discriminants. 6.3. Notions of semi-algebraic sets.