Geometric Computation (Lecture Notes Computing)

This book contains tutorial surveys and original research contributions in geometric computing, modeling, and reasoning. Highlighting the role of algebraic computation, it covers: surface blending, implicitization, and parametrization; automated deduction with Clifford algebra and in real geometry; and exact geometric computation. Basic techniques, advanced methods, and new findings are presented coherently, with many examples and illustrations. Using this book the reader will easily cross the frontiers of symbolic computation, computer aided geometric design, and automated reasoning. The book is also a valuable reference for people working in other relevant areas, such as scientific computing, computer graphics, and artificial intelligence. Algebraic methods in computer aided geometric design: theoretical and practical applications / Laureano González-Vega [und weitere] -- Constructing Piecewise algebraic blending surfaces / Yuyu Feng [und weitere] -- Rational curves and surfaces: algorithms and some applications / J. Rafael Sendra -- Panorama of methods for exact implicitization of algebraic curves and surfaces / Ilias S. Kotsireas -- Implicitization and offsetting via regular systems / Dongming Wang -- Determining the intersection curve of two 3D implicit surfaces by using differential geometry and algebraic techniques / Laureano González-Vega, Ioana Necula, and Jaime Puig-Pey -- Analytical properties of semi-stationary subdivision schemes / Hongxin Zhang and Guojin Wang -- Meshless method for numerical solution of PDE using Hermitian interpolation with radial basis / Zongmin Wu and Jianping Liu -- Clifford algebras in geometric computation / Hongbo Li -- Automated deduction in real geometry / Lu Yang and Bican Xia -- Automated derivation of unknown relations and determination of geometric loci / Yong-Bin Li -- On guaranteed accuracy computation / Chee K. Yap -- Dixon [symbol]-resultant quotients for 6-point isosceles triangular corner cutting / Mao-Ching Foo and Eng-Wee Chionh -- Face recognition using hidden Markov models and artificial neural network techniques / Zongying Ou and Bindang Xue Preface; CONTENTS; Chapter 1 Algebraic Methods in Computer Aided Geometric Design: Theoretical and Practical Applications; 1. Introduction; 2. Implicitization and Parametrization Problems; 3. Applications in CAGD; 4. Practical Performance of Algebraic Techniques in CAGD; References; Chapter 2 Constructing Piecewise Algebraic Blending Surfaces; 1. Introduction; 2. Notations and Preliminaries; 3. Direct Method; 4. Grobner Basis Method; 5. Wu's Method; 6. Syzygy Module Method; 7. Concluding Remark; Chapter 3 Rational Curves and Surfaces: Algorithms and Some Applications; 1. Introduction 8. Multidimensional Newton Formulae and Symmetric Functions9. Eigenvalue Method; 10. Conclusion; Chapter 5 Implicitization and Offsetting via Regular Systems; 1. Introduction; 2. Decomposition into Regular Systems; 3. Implicitization of Parametric Objects; 4. Computation of Offsets; 5. Simplification of Equations and Inequations; Chapter 6 Determining the Intersection Curve of Two 3D Implicit Surfaces by Using Differential Geometry and Algebraic Techniques; 1. Introduction; 2. Implicit Curves and Surfaces; 3. Intersection Problems: Sectioning; 4. Gradient Problems; 5. Examples Chapter 7 Analytical Properties of Semi-Stationary Subdivision Schemes1. Introduction; 2. Semi-stationary Schemes; 3. Convergent and Continuity Analysis; 4. Examples with Discussions; 5. Conclusion; Chapter 8 Meshless Method for Numerical Solution of PDE Using Hermitian Interpolation with Radial Basis; 1. Introduction; 2. Preliminary; 3. Solvability of the Discrete Equation; 4. The Order of the Approximation; 5. Conclusion; Chapter 9 Clifford Algebras in Geometric Computation; 1. Background; 2. Two Examples in Projective Geometry; 3. An Example in Afflne Geometry 4. First Example in Euclidean Geometry5. Second Example in Euclidean Geometry; 6. Third Example in Euclidean Geometry; 7. Further Reading; Chapter 10 Automated Deduction in Real Geometry; 1. Introduction; 2. Find Real Solutions of Geometric Problems; 3. Prove or Disprove Propositions; 4. Discover Inequality-Type Theorems; Chapter 11 Automated Derivation of Unknown Relations and Determination of Geometric Loci; 1. Introduction; 2. The Theory of Weakly Nondegenerate Conditions and Strong Regular Sets; 3. Decomposing Polynomial Systems into Strong Regular Sets 2. Algebraic Plane Curves3. Rational Plane Curves; 4. Parametrization of Rational Plane Curves; 5. Properness and Inversion; 6. Reparametrizations of Rational Plane Curves; 7. Real Rational Curves; 8. Parametrization of Rational Surfaces; 9. Some Applications; Chapter 4 Panorama of Methods for Exact Implicitization of Algebraic Curves and Surfaces; 1. Introduction; 2. Exact Implicitization of Algebraic Curves; 3. Resultants; 4. Grobner Bases; 5. Characteristic Sets; 6. Perturbations; 7. Moving Lines Moving Planes Moving Curves and Moving Surfaces