| |
| |
| Preface | |
| |
| |
| |
| The Bare Basics | |
| |
| |
| |
| Points and Vectors | |
| |
| |
| |
| Operations | |
| |
| |
| |
| Products | |
| |
| |
| |
| Affine Maps | |
| |
| |
| |
| Triangles and Tetrahedra | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Lines and Planes | |
| |
| |
| |
| Linear Interpolation | |
| |
| |
| |
| Line Forms | |
| |
| |
| |
| Planes | |
| |
| |
| |
| Linear Pieces: Polygons | |
| |
| |
| |
| Linear Pieces: Triangulations | |
| |
| |
| |
| Working with Triangulations | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Cubic Bezier Curves | |
| |
| |
| |
| Parametric Curves | |
| |
| |
| |
| Cubic Bezier Curves | |
| |
| |
| |
| Derivatives | |
| |
| |
| |
| The de Casteljau Algorithm | |
| |
| |
| |
| Subdivision | |
| |
| |
| |
| Exploring the Properties of Bezier Curves | |
| |
| |
| |
| The Matrix Form and Monomials | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Bezier Curves: Cubic and Beyond | |
| |
| |
| |
| Bezier Curves | |
| |
| |
| |
| Derivatives Revisited | |
| |
| |
| |
| The de Casteljau Algorithm Revisited | |
| |
| |
| |
| The Matrix Form and Monomials Revisited | |
| |
| |
| |
| Degree Elevation | |
| |
| |
| |
| Degree Reduction | |
| |
| |
| |
| Bezier Curves over General Intervals | |
| |
| |
| |
| Functional Bezier Curves | |
| |
| |
| |
| More on Bernstein Polynomials | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Putting Curves to Work | |
| |
| |
| |
| Cubic Interpolation | |
| |
| |
| |
| Interpolation Beyond Cubics | |
| |
| |
| |
| Aitken's Algorithm | |
| |
| |
| |
| Approximation | |
| |
| |
| |
| Finding the Right Parameters | |
| |
| |
| |
| Hermite Interpolation | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Bezier Patches | |
| |
| |
| |
| Parametric Surfaces | |
| |
| |
| |
| Bilinear Patches | |
| |
| |
| |
| Bezier Patches | |
| |
| |
| |
| Properties of Bezier Patches | |
| |
| |
| |
| Derivatives | |
| |
| |
| |
| Higher Order Derivatives | |
| |
| |
| |
| The de Casteljau Algorithm | |
| |
| |
| |
| Normals | |
| |
| |
| |
| Changing Degrees | |
| |
| |
| |
| Subdivision | |
| |
| |
| |
| Ruled Bezier Patches | |
| |
| |
| |
| Functional Bezier Patches | |
| |
| |
| |
| Monomial Patches | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Working with Polynomial Patches | |
| |
| |
| |
| Bicubic Interpolation | |
| |
| |
| |
| Interpolation using Higher Degrees | |
| |
| |
| |
| Coons Patches | |
| |
| |
| |
| Bicubic Hermite Interpolation | |
| |
| |
| |
| Trimmed Patches | |
| |
| |
| |
| Least Squares Approximation | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Shape | |
| |
| |
| |
| The Frenet Frame | |
| |
| |
| |
| Curvature and Torsion | |
| |
| |
| |
| Surface Curvatures | |
| |
| |
| |
| Reflection Lines | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Composite Curves | |
| |
| |
| |
| Piecewise Bezier Curves | |
| |
| |
| |
| C[superscript 1] and G[superscript 1] Continuity | |
| |
| |
| |
| C[superscript 2] and G[superscript 2] Continuity | |
| |
| |
| |
| Working with Piecewise Bezier Curves | |
| |
| |
| |
| Point-Normal Interpolation | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| B-Spline Curves | |
| |
| |
| |
| Basic Definitions | |
| |
| |
| |
| The de Boor Algorithm | |
| |
| |
| |
| Practicalities of the de Boor Algorithm | |
| |
| |
| |
| Properties of B-Spline Curves | |
| |
| |
| |
| B-Splines: The Building Block | |
| |
| |
| |
| Knot Insertion | |
| |
| |
| |
| Periodic B-Spline Curves | |
| |
| |
| |
| Derivatives | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Working with B-Spline Curves | |
| |
| |
| |
| Designing with B-Spline Curves | |
| |
| |
| |
| Least Squares Approximation | |
| |
| |
| |
| Shape Equations | |
| |
| |
| |
| Cubic Spline Interpolation | |
| |
| |
| |
| Cubic Spline Interpolation in a Nutshell | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Composite Surfaces | |
| |
| |
| |
| Composite Bezier Surfaces | |
| |
| |
| |
| B-Spline Surfaces | |
| |
| |
| |
| B-Spline Surface Approximation | |
| |
| |
| |
| B-Spline Surface Interpolation | |
| |
| |
| |
| Subdivision Surfaces: Doo-Sabin | |
| |
| |
| |
| Subdivision Surfaces: Catmull-Clark | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Nurbs | |
| |
| |
| |
| Conics | |
| |
| |
| |
| Reparametrization and Classification | |
| |
| |
| |
| Derivatives | |
| |
| |
| |
| The Circle | |
| |
| |
| |
| Rational Bezier Curves | |
| |
| |
| |
| Rational B-Spline Curves | |
| |
| |
| |
| Rational Bezier and B-spline Surfaces | |
| |
| |
| |
| Surfaces of Revolution | |
| |
| |
| |
| Exercises | |
| |
| |
| |
| Hunting Geometry Bugs | |
| |
| |
| Solutions to Selected Exercises | |
| |
| |
| Notation | |
| |
| |
| Bibliography | |
| |
| |
| Index | |