Polyhedral And Algebraic Methods In Computational Geometry 2013th Edition Michael Joswig by Michael Joswig, Thorsten Theobald 9781447148166, 1447148169 instant download after payment.

Polyhedral and Algebraic Methods in Computational Geometry provides a thorough introduction into algorithmic geometry and its applications. It presents its primary topics from the viewpoints of discrete, convex and elementary algebraic geometry. The first part of the book studies classical problems and techniques that refer to polyhedral structures. The authors include a study on algorithms for computing convex hulls as well as the construction of Voronoi diagrams and Delone triangulations. The second part of the book develops the primary concepts of (non-linear) computational algebraic geometry. Here, the book looks at Gröbner bases and solving systems of polynomial equations. The theory is illustrated by applications in computer graphics, curve reconstruction and robotics. Throughout the book, interconnections between computational geometry and other disciplines (such as algebraic geometry, optimization and numerical mathematics) are established. Polyhedral and Algebraic Methods in Computational Geometry is directed towards advanced undergraduates in mathematics and computer science, as well as towards engineering students who are interested in the applications of computational geometry.
Table of Contents
Cover
Polyhedral and Algebraic Methods in Computational Geometry
ISBN 9781447148166 ISBN 9781447148173
Preface
Contents
Introduction and Overview
1.1 Linear Computational Geometry
1.2 Non-linear Computational Geometry
1.3 Applications
Part I Linear Computational Geometry
Geometric Fundamentals
2.1 Projective Spaces
2.2 Projective Transformations
2.3 Convexity
o 2.3.1 Orientation of Affin Hyperplanes
o 2.3.2 Separation Theorems
2.4 Exercises
2.5 Remarks
Polytopes and Polyhedra
3.1 Definition and Fundamental Properties
o 3.1.1 The Faces of a Polytope
o 3.1.2 First Consequences of the Separating Hyperplane Theorem
o 3.1.3 The Outer Description of a Polytope
3.2 The Face Lattice of a Polytope
3.3 Polarity and Duality
3.4 Polyhedra
3.5 The Combinatorics of Polytopes
3.6 Inspection Using polymake
o 3.6.1 Cyclic Polytopes
o 3.6.2 Random Polytopes
o 3.6.3 Projective Transformations
3.7 Exercises
3.8 Remarks
Linear Programming
4.1 The Task
4.2 Duality
4.3 The Simplex Algorithm
4.4 Determining a Start Vertex
4.5 Inspection Using polymake
4.6 Exercises
4.7 Remarks
Computation of Convex Hulls
5.1 Preliminary Considerations
5.2 The Double Description Method
5.3 Convex Hulls in the Plane
5.4 Inspection Using polymake
5.5 Exercises
5.6 Remarks
Voronoi Diagrams
6.1 Voronoi Regions
6.2 Polyhedral Complexes
6.3 Voronoi Diagrams and Convex Hulls
6.4 The Beach Line Algorithm
o 6.4.1 The Algorithm
6.5 Determining the Nearest Neighbor
6.6 Exercises
6.7 Remarks
Delone Triangulations
7.1 Duality of Voronoi Diagrams
7.2 The Delone Subdivision
7.3 Computation of Volumes
7.4 Optimality of Delone Triangulations
7.5 Planar Delone Triangulations
7.6 Inspection Using polymake
7.7 Exercises
7.8 Remarks
Part II Non-linear Computational Geometry
Algebraic and Geometric Foundations
8.1 Motivation
8.2 Univariate Polynomials
8.3 Resultants
8.4 Plane Affin Algebraic Curves
8.5 Projective Curves
8.6 B�zout's Theorem
8.7 Algebraic Curves Using Maple
8.8 Exercises
8.9 Remarks
Gr�bner Bases and Buchberger's Algorithm
9.1 Ideals and the Univariate Case
9.2 Monomial Orders
9.3 Gr�bner Bases and the Hilbert Basis Theorem
9.4 Buchberger's Algorithm
9.5 Binomial Ideals
9.6 Proving a Simple Geometric Fact Using Gr�bner Bases
9.7 Exercises
9.8 Remarks
Solving Systems of Polynomial Equations Using Gr�bner Bases
10.1 Gr�bner Bases Using Maple and Singular
10.2 Elimination of Unknowns
10.3 Continuation of Partial Solutions
10.4 The Nullstellensatz
10.5 Solving Systems of Polynomial Equations
10.6 Gr�bner Bases and Integer Linear Programs
10.7 Exercises
10.8 Remarks
Part III Applications
Reconstruction of Curves
11.1 Preliminary Considerations
11.2 Medial Axis and Local Feature Size
11.3 Samples and Polygonal Reconstruction
11.4 The Algorithm NN-Crust
11.5 Curve Reconstruction with polymake
11.6 Exercises
11.7 Remarks
Pl�cker Coordinates and Lines in Space
12.1 Pl�cker Coordinates
12.2 Exterior Multiplication and Exterior Algebra
12.3 Duality
12.4 Computations with Pl�cker Coordinates
12.5 Lines in R3
o 12.5.1 Transversals
12.6 Exercises
12.7 Remarks
Applications of Non-linear Computational Geometry
13.1 Voronoi Diagrams for Line Segments in the Plane
13.2 Kinematic Problems and Motion Planning
13.3 The Global Positioning System GPS
13.4 Exercises
13.5 Remarks
Algebraic Structures
Separation Theorems
Algorithms and Complexity
Software
Notation
References
Index