Curves and Surfaces in Geometric Modeling
Theory & Algorithms
- 1st Edition - October 7, 1999
- Author: Jean Gallier
- Language: English
- Hardback ISBN:9 7 8 - 1 - 5 5 8 6 0 - 5 9 9 - 2
- eBook ISBN:9 7 8 - 0 - 0 8 - 0 5 0 3 5 3 - 0
Curves and Surfaces for Geometric Design offers both a theoretically unifying understanding of polynomial curves and surfaces and an effective approach to implementation that you… Read more
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offers both a theoretically unifying understanding of polynomial curves and surfaces and an effective approach to implementation that you can bring to bear on your own work—whether you're a graduate student, scientist, or practitioner.Inside, the focus is on "blossoming"—the process of converting a polynomial to its polar form—as a natural, purely geometric explanation of the behavior of curves and surfaces. This insight is important for far more than its theoretical elegance, for the author proceeds to demonstrate the value of blossoming as a practical algorithmic tool for generating and manipulating curves and surfaces that meet many different criteria. You'll learn to use this and related techniques drawn from affine geometry for computing and adjusting control points, deriving the continuity conditions for splines, creating subdivision surfaces, and more.
The product of groundbreaking research by a noteworthy computer scientist and mathematician, this book is destined to emerge as a classic work on this complex subject. It will be an essential acquisition for readers in many different areas, including computer graphics and animation, robotics, virtual reality, geometric modeling and design, medical imaging, computer vision, and motion planning.
* Achieves a depth of coverage not found in any other book in this field.* Offers a mathematically rigorous, unifying approach to the algorithmic generation and manipulation of curves and surfaces. * Covers basic concepts of affine geometry, the ideal framework for dealing with curves and surfaces in terms of control points.* Details (in Mathematica) many complete implementations, explaining how they produce highly continuous curves and surfaces.* Presents the primary techniques for creating and analyzing the convergence of subdivision surfaces (Doo-Sabin, Catmull-Clark, Loop).* Contains appendices on linear algebra, basic topology, and differential calculus.
Computer scientists (students and professionals), mathematicians, and engineers interested in geometric methods
1 Introduction1.1 Geometric Methods in Engineering1.2 Examples of Problems Using Geometric Modeling2 Basics of Affine Geometry 2.1 Affine Spaces2.2 Examples of Affine Spaces2.3 Chasles' Identity2.4 Affine Combinations, Barycenters2.5 Affine Subspaces2.6 Affine Independence and Affine Frames2.7 Affine Maps2.8 Affine Groups2.9 Affine Hyperplanes3 Introduction to the Algorithmic Geometry of Polynomial Curves3.1 Why Parameterized Polynomial Curves?3.2 Polynomial Curves of degree 1 and 23.3 First Encounter with Polar Forms (Blossoming)3.4 First Encounter with the de Casteljau Algorithm3.5 Polynomial Curves of Degree 33.6 Classification of the Polynomial Cubics3.7 Second Encounter with Polar Forms (Blossoming)3.8 Second Encounter with the de Casteljau Algorith3.9 Examples of Cubics Defined by Control Points4 Multiaffine Maps and Polar Forms4.1 Multiaffine Maps4.2 Affine Polynomials and Polar Forms4.3 Polynomial Curves and Control Points4.4 Uniqueness of the Polar Form of an Affine Polynomial Map4.5 Polarizing Polynomials in One or Several Variables5 Polynomial Curves as Be'zier Curves5.1 The de Casteljau Algorithm5.2 Subdivision Algorithms for Polynomial Curves5.3 The Progressive Version of the de Casteljau Algorithm (the de Boor Algorithm)5.4 Derivatives of Polynomial Curves5.5 Joining Affine Polynomial Functions6 B-Spline Curves6.1 Introduction: Knot Sequences, de Boor Control Points6.2 Infinite Knot Sequences, Open B-Spline Curves6.3 Finite Knot Sequences, Finite B-Spline Curves6.4 Cyclic Knot Sequences, Closed (Cyclic) B-Spline Curves6.5 The de Boor Algorithm6.6 The de Boor Algorithm and Knot Insertion6.7 Polar forms of B-Splines6.8 Cubic Spline Interpolation7 Polynomial Surfaces7.1 Polarizing Polynomial Surfaces7.2 Bipolynomial Surfaces in Polar Form7.3 The de Casteljau Algorithm for Rectangular Surface Patches7.4 Total Degree Surfaces in Polar Form7.5 The de Casteljau Algorithm for Triangular Surface Patches7.6 Directional Derivatives of Polynomial Surfaces8 Subdivision Algorithms for Polynomial Surfaces 2948.1 Subdivision Algorithms for Triangular Patches8.2 Subdivision Algorithms for Rectangular Patches9. Polynomial Spline Surfaces and Subdivision Surfaces9.1 Joining Polynomial Surfaces9.2 Spline Surfaces with Triangular Patches9.3 Spline Surfaces with Rectangular Patches9.4 Subdivision Surfaces10 Embedding an Affine Space in a Vector Space 36910.1 The "Hat Construction", or Homogenizing10.2 Affine Frames of E and Bases of E10.3 Extending Affine Maps to Linear Maps10.4 From Multiaffine Maps to Multilinear Maps10.5 Differentiating Affine Polynomial Functions Using Their Homogenized Polar Forms, Osculating Flats11 Tensor Products and Symmetric Tensor Products 39511.1 Tensor Products11.2 Symmetric Tensor Products11.3 Affine Symmetric Tensor Products11.4 Properties of Symmetric Tensor Products11.5 Polar Forms Revisited12 Appendix 1: Linear Algebra12.1 Vector Spaces12.2 Linear Maps12.3 Quotient Spaces12.4 Direct Sums12.5 Hyperplanes and Linear Forms13 Appendix 2: Complements of Affine Geometry 44213.1 Affine and Multiaffine Maps13.2 Homogenizing Multiaffine Maps13.3 Intersection and Direct Sums of Affine Spaces13.4 Osculating Flats Revisited14 Appendix 3: Topology14.1 Metric Spaces and Normed Vector Spaces14.2 Continuous Functions, Limits14.3 Normed Affine Spaces 46515 Appendix 4: Differential Calculus15.1 Directional Derivatives, Total Derivatives15.2 Jacobian Matrices
- No. of pages: 491
- Language: English
- Edition: 1
- Published: October 7, 1999
- Imprint: Morgan Kaufmann
- Hardback ISBN: 9781558605992
- eBook ISBN: 9780080503530
JG
Jean Gallier
Jean Gallier received the degree of Civil Engineer from the Ecole Nationale des Ponts et Chaussees in 1972 and a Ph.D. in Computer Science from UCLA in 1978. That same year he joined the University of Pennsylvania, where he is presently a professor in CIS with a secondary appointment in Mathematics. In 1983, he received the Linback Award for distinguished teaching. Gallier’s research interests range from constructive logics and automated theorem proving to geometry and its applications to computer graphics, animation, computer vision, and motion planning. The author of Logic in Computer Science, he enjoys hiking (especially the Alps) and swimming. He also enjoys classical music (Mozart), jazz (Duke Ellington, Oscar Peterson), and wines from Burgundy, especially Volnay.