book

Pyramid Algorithms

by Ron Goldman

July 2002

Intermediate to advanced

576 pages

15h 37m

English

Morgan Kaufmann

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1.1 Ambient Spaces1.2 Coordinates1.3 Curve and Surface Representations1.4 Summary

2.1 Linear Interpolation2.2 Neville’s Algorithm2.3 The Structure of Neville’s Algorithm2.4 Uniqueness of Polynomial Interpolants and Taylor’s Theorem2.5 Lagrange Basis Functions2.6 Computational Techniques for Lagrange Interpolation2.7 Rational Lagrange Curves2.8 Fast Fourier Transform2.9 Recapitulation2.10 Surface Interpolation2.11 Rectangular Tensor Product Lagrange Surfaces2.12 Triangular Lagrange Patches2.13 Uniqueness of the Bivariate Lagrange Interpolant2.14 Rational Lagrange Surfaces2.15 Ruled, Lofted, and Boolean Sum Surfaces2.16 Summary
3.1 Cubic Hermite Interpolation3.2 Neville’s Algorithm for General Hermite Interpolation3.3 The Hermite Basis Functions3.4 Rational Hermite Curves3.5 Hermite Surfaces3.6 Summary
4.1 The Newton Basis4.2 Divided Differences4.3 Properties of Divided Differences4.4 An Axiomatic Approach to Divided Difference4.5 Forward Differencing4.6 Summary
5.1 De Casteljau’s Algorithm5.2 Elementary Properties of Bezier Curves5.3 The Bernstein Basis Functions and Pascal’s Triangle5.4 More Properties of Bernstein/Bezier Curves5.5 Change of Basis Procedures and Principles of Duality
5.7 Rational Bezier Curves5.8 Bezier Surfaces5.9 Summary
6.1 Blossoming the de Casteljau Algorithm6.2 Existence and Uniqueness of the Blossom6.3 Change of Basis Algorithms6.4 Differentiation and the Homogeneous Blossom6.5 Blossoming Bezier Patches6.6 Summary
7.1 The de Boor Algorithm7.2 Progressive Polynomial Bases Generated by Progressive Knot Sequences7.3 B-Spline Curves7.4 Elementary Properties of B-Spline Curves7.5 All Splines Are B-Splines7.6 Knot Insertion Algorithms7.7 The B-Spline Basis Functions
7.9 Rational B-Splines7.10 Catmull-Rom Splines7.11 Tensor Product B-Spline Surfaces7.12 Pyramid Algorithms and Triangular B-Patches7.13 Summary
8.1 Barycentric Coordinates for Convex Polygons8.2 Polygonal Arrays8.3 Neville’s Pyramid Algorithm and Multisided Grids8.4 S-Patches8.5 Pyramid Patches and the General Pyramid Algorithm8.6 C-Patches
8.8 Summary

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About the Author

Ron Goldman has been a professor of computer science at Rice University since 1990. Previously he worked in industry, solving problems in computer graphics, geometric modeling, and computer-aided design. He served as a mathematician at Manufacturing Data Systems Inc., where he helped implement one of the first industrial solid-modeling systems. Later he was employed as a senior design engineer at Ford Motor Company, enhancing the capabilities of their corporate graphics and computer-aided design software. From Ford, Dr. Goldman moved to Control Data Corporation as a principal consultant for the development group devoted to computer-aided design and manufacturing. He left Control Data Corporation in 1987 to become an associate ...