COMPUTING IN EUCLIDEAN GEOMETRY (V1)

By: DU D Z

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Hardcover | 11 September 1992 | Edition Number 2

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398 Pages

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22.86 x 15.24 x 3.18

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This book is a collection of surveys and exploratory articles about recent developments in the field of computational Euclidean geometry. The topics covered are: a history of Euclidean geometry, Voronoi diagrams, randomized geometric algorithms, computational algebra; triangulations, machine proofs, topological designs, finite-element mesh, computer-aided geometric designs and steiner trees. Each chapter is written by a leading expert in the field and together they provide a clear and authoritative picture of what computational Euclidean geometry is and the direction in which research is going.

Industry Reviews

"D-Z Du and F Hwang have put to rest an optimization problem known as the Steiner ratio conjecture. Their solution closes the book on a problem that had frustrated a generation of geometers, but it also writes the first chapter of a new volume. The key to Du and Hwang's successful attack on the conjecture is a new method that has potential for solving a raft of other optimization problems." Barry A Cipra SIAM News, USA "... the eight surveys are well organized. Each survey is preceded by a good introductory section with a rich bibliography. Both beginners and experts will benefit from this book." Jie Tian Mathematical Reviews "The papers are not just summaries; the authors present new material or fresh points of view ... I recommend the book to anyone who works in one of the areas surveyed or who is interested in the interaction of Euclidean geometry and computers." Carol Hazlewood IEEE Parallel & Distributed Technology

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Non-Fiction Mathematics Geometry Euclidean Geometry Computing & I.T.Computer Science Mathematical Theory of Computation Booktopia Publisher Services World Scientific Publishing

Preface
On the Development of Quantitative Geometry from Pythagoras to Grassmann	p. 1
Introduction	p. 1
Foundation of Quantitative Geometry	p. 2
Commensurability and non-commensurability	p. 3
The area formula of rectangles, Pythagoras theorem and the similar triangle theorem	p. 3
Euclidean trigonometry	p. 5
Foundation of Analytic (Vector) Geometry	p. 6
Displacement vectors	p. 6
Oriented volume and high dimensional inner product	p. 10
The Grassmann algebra	p. 12
Some Applications	p. 15
Spherical trigonometry and spherical geometry	p. 15
A volume formula of superscribing polyhedrons	p. 19
References	p. 20
Mesh Generation and Optimal Triangulation	p. 23
Introduction	p. 23
Background	p. 24
Formulating the problems	p. 25
Organization	p. 26
Two-dimensional Triangulations	p. 27
Triangulation without optimization	p. 28
Optimal triangulation	p. 29
Steiner triangulation	p. 43
Heuristically generated meshes	p. 57
Two-and-a-half-dimensional problems	p. 63
Three-dimensional Triangulations	p. 68
Tetrahedralization without optimization	p. 68
Optimal tetrahedralization	p. 72
Steiner tetrahedralization	p. 75
Heuristically generated three-dimensional meshes	p. 78
Conclusions	p. 80
References	p. 80
Machine Proofs of Geometry Theorems	p. 91
Introduction	p. 91
History of Automated Theorem Proving	p. 92
Traditional Proofs	p. 94
Algebraic Proofs	p. 96
Wu's Method and Its Variants	p. 97
Grobner basis method	p. 105
Algebraic v/s Traditional Proofs	p. 108
Axiomatic geometries and number systems	p. 108
Algebraic proofs v/s formal proofs	p. 111
Concluding Remarks	p. 111
References	p. 112
Randomized Geometric Algorithms	p. 117
Introduction	p. 117
Outline	p. 118
Probabilistic Divide-and-Conquer	p. 119
Closest-point queries	p. 119
The Vapnik-Chervonenkis dimension	p. 123
Sharper Expected Bounds for Divide-and-Conquer	p. 126
Trapezoidal diagrams of simple polygons	p. 126
Proving the expected bounds	p. 129
General segments and lines	p. 131
Randomized Incremental Algorithms	p. 132
Convex hulls	p. 132
Analysis of insertions	p. 135
Other search schemes	p. 138
Trapezoidal diagrams and planar point location	p. 140
Some Combinatorial Bounds	p. 141
Incidences	p. 141
Bounds on [actual symbol not reproducible]-regions	p. 144
Still Sharper Bounds for Divide-and-Conquer	p. 147
Bounds for the sum of squares	p. 148
Uniform bounds and cuttings	p. 149
Reweighting Methods	p. 150
Linear programming	p. 150
Triangular partitions	p. 152
Closing Remarks	p. 154
Appendix A. Some Probability, Geometry, and Notation	p. 154
Appendix B.	p. 155
References	p. 157
The State of Art on Steiner Ratio Problems	p. 163
Introduction	p. 163
Background of the [actual symbol not reproducible] Conjecture	p. 164
A Minimax Theorem	p. 166
Four Points	p. 169
Minimum Hexagonal Trees	p. 172
Inner Spanning Trees	p. 174
A Final Touch for a Rigorous Proof	p. 179
A Background of the Better Heuristics Problem	p. 181
A Better Heuristic for an Arbitrary Metric Space	p. 182
Lower Bounds of Q[subscript k]	p. 184
Lower Bounds of [rho [subscript k]] for the Rectilinear Metric	p. 186
Open Problems	p. 188
References	p. 189
Voronoi Diagrams and Delaunay Triangulations	p. 193
Introduction	p. 193
Definition of the Voronoi Diagram and Delaunay Triangulation	p. 197
Voronoi diagrams and Delaunay triangulations	p. 197
Connection with convex polyhedra	p. 200
Combinatorial complexity	p. 203
Properties of the Voronoi Diagram and Delaunay Triangulation	p. 205
Optimality of the Delaunay triangulation	p. 205
Geometric graph properties	p. 206
Inscribability	p. 207
Algorithms	p. 209
Primitives for Delaunay triangulation algorithms	p. 210
Flipping	p. 212
The incremental algorithm	p. 214
The random incremental algorithm	p. 217
The plane-sweep algorithm	p. 220
Other algorithms	p. 223
The real RAM and the general position assumption	p. 223
Implementation issues	p. 226
Appendix 1. Definitions from the Theory of Polyhedra	p. 228
Appendix 2. Proof of Theorem 2.1	p. 229
References	p. 230
Polar Forms and Triangular B-Spline Surfaces	p. 235
Introduction	p. 235
Polynomials and Polar Forms	p. 237
Triangular Bezier Patches	p. 242
The Bernstein-Bezier Representation	p. 242
The de Casteljau algorithm and its applications	p. 247
B-Patches	p. 255
The B-patch representation	p. 255
The de Boor algorithm and its applications	p. 257
Bivariate B-Splines	p. 263
A New B-Spline Scheme	p. 267
A new spline space	p. 267
Representing piecewise polynomials as linear combinations of B-splines	p. 274
Properties of the new B-spline scheme	p. 277
References	p. 281
Computational Geometry and Topological Network Design	p. 287
Introduction	p. 288
Purpose and focus	p. 288
Overview of the survey	p. 288
Network Design Problems	p. 289
Topological network design	p. 291
Routing network design	p. 292
Capacitated network design	p. 292
Geometric topological network design	p. 293
TND definitions and assumptions	p. 293
Computational Geometry	p. 294
Mathematical programming vs. geometric approaches	p. 296
Combinatorial optimization vs. geometric approaches	p. 298
Easy vs. difficult problems	p. 298
Paradigms and Geometric Data Structures	p. 299
Paradigms	p. 299
Geometric data structures	p. 305
Geometric TND Problems	p. 319
G(Z) problems	p. 320
G(E) problems	p. 333
G(Z,E) problems	p. 344
G(Omega) problems	p. 353
[Actual symbol not reproducible] cube	p. 361
Composite matrix	p. 365
Planar graph diagram	p. 365
[Actual symbol not reproducible] decomposition	p. 367
Open problems	p. 369
Summary and Conclusions	p. 369
References	p. 371
Table of Contents provided by Blackwell. All Rights Reserved.

COMPUTING IN EUCLIDEAN GEOMETRY (V1)

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