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E-grāmata: Polyhedral and Algebraic Methods in Computational Geometry

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  • Formāts: PDF+DRM
  • Sērija : Universitext
  • Izdošanas datums: 04-Jan-2013
  • Izdevniecība: Springer London Ltd
  • Valoda: eng
  • ISBN-13: 9781447148173
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Polyhedral and Algebraic Methods in Computational GeometryPalielināt
  • Formāts: PDF+DRM
  • Sērija : Universitext
  • Izdošanas datums: 04-Jan-2013
  • Izdevniecība: Springer London Ltd
  • Valoda: eng
  • ISBN-13: 9781447148173
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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.

Recenzijas

From the reviews:

The authors discuss in the book a selection of linear and non-linear topics in computational geometry. The books audience is made up of mathematicians interested in applications of geometry and algebra as well as computer scientists and engineers with good mathematical background. (Antonio Valdés Morales, The European Mathematical Society, September, 2013)

1 Introduction and Overview
1(8)
1.1 Linear Computational Geometry
1(3)
1.2 Non-linear Computational Geometry
4(1)
1.3 Applications
5(4)
Appendix
6(3)
Part I Linear Computational Geometry
2 Geometric Fundamentals
9(10)
2.1 Projective Spaces
9(3)
2.2 Projective Transformations
12(1)
2.3 Convexity
13(3)
2.4 Exercises
16(1)
2.5 Remarks
17(2)
3 Polytopes and Polyhedra
19(28)
3.1 Definitions and Fundamental Properties
19(6)
3.2 The Face Lattice of a Polytope
25(3)
3.3 Polarity and Duality
28(3)
3.4 Polyhedra
31(3)
3.5 The Combinatorics of Polytopes
34(6)
3.6 Inspection Using polymake
40(4)
3.7 Exercises
44(1)
3.8 Remarks
45(2)
4 Linear Programming
47(18)
4.1 The Task
47(2)
4.2 Duality
49(4)
4.3 The Simplex Algorithm
53(7)
4.4 Determining a Start Vertex
60(1)
4.5 Inspection Using polymake
61(2)
4.6 Exercises
63(1)
4.7 Remarks
64(1)
5 Computation of Convex Hulls
65(16)
5.1 Preliminary Considerations
65(1)
5.2 The Double Description Method
66(6)
5.3 Convex Hulls in the Plane
72(4)
5.4 Inspection Using polymake
76(1)
5.5 Exercises
77(1)
5.6 Remarks
78(3)
6 Voronoi Diagrams
81(18)
6.1 Voronoi Regions
81(2)
6.2 Polyhedral Complexes
83(1)
6.3 Voronoi Diagrams and Convex Hulls
84(4)
6.4 The Beach Line Algorithm
88(8)
6.5 Determining the Nearest Neighbor
96(1)
6.6 Exercises
97(1)
6.7 Remarks
98(1)
7 Delone Triangulations
99(20)
7.1 Duality of Voronoi Diagrams
99(3)
7.2 The Delone Subdivision
102(2)
7.3 Computation of Volumes
104(1)
7.4 Optimality of Delone Triangulations
105(4)
7.5 Planar Delone Triangulations
109(5)
7.6 Inspection Using polymake
114(2)
7.7 Exercises
116(1)
7.8 Remarks
116(3)
Part II Non-linear Computational Geometry
8 Algebraic and Geometric Foundations
119(18)
8.1 Motivation
119(3)
8.2 Univariate Polynomials
122(1)
8.3 Resultants
123(2)
8.4 Plane Affine Algebraic Curves
125(2)
8.5 Projective Curves
127(2)
8.6 Bezout's Theorem
129(4)
8.7 Algebraic Curves Using Maple
133(2)
8.8 Exercises
135(1)
8.9 Remarks
136(1)
9 Grobner Bases and Buchberger's Algorithm
137(20)
9.1 Ideals and the Univariate Case
137(4)
9.2 Monomial Orders
141(4)
9.3 Grobner Bases and the Hilbert Basis Theorem
145(4)
9.4 Buchberger's Algorithm
149(3)
9.5 Binomial Ideals
152(1)
9.6 Proving a Simple Geometric Fact Using Grobner Bases
153(2)
9.7 Exercises
155(1)
9.8 Remarks
155(2)
10 Solving Systems of Polynomial Equations Using Grobner Bases
157(24)
10.1 Grobner Bases Using Maple and Singular
157(1)
10.2 Elimination of Unknowns
158(4)
10.3 Continuation of Partial Solutions
162(2)
10.4 The Nullstellensatz
164(3)
10.5 Solving Systems of Polynomial Equations
167(4)
10.6 Grobner Bases and Integer Linear Programs
171(6)
10.7 Exercises
177(1)
10.8 Remarks
177(4)
Part III Applications
11 Reconstruction of Curves
181(12)
11.1 Preliminary Considerations
181(1)
11.2 Medial Axis and Local Feature Size
182(3)
11.3 Samples and Polygonal Reconstruction
185(2)
11.4 The Algorithm NN-Crust
187(3)
11.5 Curve Reconstruction with polymake
190(1)
11.6 Exercises
190(2)
11.7 Remarks
192(1)
12 Plucker Coordinates and Lines in Space
193(16)
12.1 Plucker Coordinates
193(1)
12.2 Exterior Multiplication and Exterior Algebra
194(5)
12.3 Duality
199(4)
12.4 Computations with Plucker Coordinates
203(1)
12.5 Lines in R3
204(2)
12.6 Exercises
206(1)
12.7 Remarks
206(3)
13 Applications of Non-linear Computational Geometry
209(14)
13.1 Voronoi Diagrams for Line Segments in the Plane
209(3)
13.2 Kinematic Problems and Motion Planning
212(7)
13.3 The Global Positioning System GPS
219(2)
13.4 Exercises
221(1)
13.5 Remarks
222(1)
Appendix A Algebraic Structures
223(4)
A.1 Groups, Rings, Fields
223(1)
A.2 Polynomial Rings
224(3)
Appendix B Separation Theorems
227(4)
Appendix C Algorithms and Complexity
231(6)
C.1 Complexity of Algorithms
231(2)
C.2 The Complexity Classes P and NP
233(4)
Appendix D Software
237(4)
D.1 Polymake
237(1)
D.2 Maple
237(1)
D.3 Singular
238(1)
D.4 CGAL
238(1)
D.5 Sage
238(3)
Appendix E Notation
241(2)
References 243(4)
Index 247
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