Barycentric Calculus In Euclidean And Hyperbolic Geometry: A Comparative Introduction

by Abraham Albert Ungar

Overview

The word barycentric is derived from the Greek word barys (heavy), and refers to center of gravity. Barycentric calculus is a method of treating geometry by considering a point as the center of gravity of certain other points to which weights are ascribed. Hence, in particular, barycentric calculus provides excellent insight into triangle centers. This unique book on barycentric calculus in Euclidean and hyperbolic geometry provides an introduction to the fascinating and beautiful subject of novel triangle centers in hyperbolic geometry along with analogies they share with familiar triangle centers in Euclidean geometry. As such, the book uncovers magnificent unifying notions that Euclidean and hyperbolic triangle centers share.In his earlier books the author adopted Cartesian coordinates, trigonometry and vector algebra for use in hyperbolic geometry that is fully analogous to the common use of Cartesian coordinates, trigonometry and vector algebra in Euclidean geometry. As a result, powerful tools that are commonly available in Euclidean geometry became available in hyperbolic geometry as well, enabling one to explore hyperbolic geometry in novel ways. In particular, this new book establishes hyperbolic barycentric coordinates that are used to determine various hyperbolic triangle centers just as Euclidean barycentric coordinates are commonly used to determine various Euclidean triangle centers.The hunt for Euclidean triangle centers is an old tradition in Euclidean geometry, resulting in a repertoire of more than three thousand triangle centers that are known by their barycentric coordinate representations. The aim of this book is to initiate a fully analogous hunt for hyperbolic triangle centers that will broaden the repertoire of hyperbolic triangle centers provided here.

Product Details

ISBN-13:	9789814304931
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	08/27/2010
Pages:	360
Product dimensions:	6.10(w) x 9.00(h) x 1.00(d)

Table of Contents

Preface vii

1 Euclidean Barycentric Coordinates and the Classic Triangle Centers 1

1.1 Points, Lines, Distance and Isometries 2

1.2 Vectors, Angles and Triangles 5

1.3 Euclidean Barycentric Coordinates 8

1.4 Analogies with Classical Mechanics 11

1.5 Barycentric Representations are Covariant 12

1.6 Vector Barycentric Representation 14

1.7 Triangle Centroid 17

1.8 Triangle Altitude 19

1.9 Triangle Orthocenter 24

1.10 Triangle Incenter 27

1.11 Triangle Inradius 33

1.12 Triangle Circumcenter 36

1.13 Circumradius 40

1.14 Triangle Incircle and Excircles 42

1.15 Excircle Tangency Points 47

1.16 From Triangle Tangency Points to Triangle Centers 52

1.17 Triangle In-Exradii 55

1.18 A Step Toward the Comparative Study 57

1.19 Tetrahedron Altitude 58

1.20 Tetrahedron Altitude Length 62

1.21 Exercises 63

2 Gyrovector Spaces and Cartesian Models of Hyperbolic Geometry 65

2.1 Einstein Addition 66

2.2 Einstein Gyration 70

2.3 From Einstein Velocity Addition to Gyrogroups 73

2.4 First Gyrogroup Theorems 77

2.5 The Two Basic Equations of Gyrogroups 82

2.6 Einstein Gyrovector Spaces 86

2.7 Gyrovector Spaces 89

2.8 Einstein Points, Gyrolines and Gyrodistance 95

2.9 Linking Einstein Addition to Hyperbolic Geometry 99

2.10 Einstein Gyrovectors, Gyroangles and Gyrotriangles 101

2.11 The Law of Gyrocosines 106

2.12 The SSS to AAA Conversion Law 108

2.13 Inequalities for Gyrotriangles 109

2.14 The AAA to SSS Conversion Law 111

2.15 The Law of Gyrosines 115

2.16 The ASA to SAS Conversion Law 115

2.17 Gyrotriangle Defect 116

2.18 Right Gyrotriangles 118

2.19 Einstein Gyrotrigonometry and Gyroarea 120

2.20 Gyrotriangle Gyroarea Addition Law 124

2.21 Gyrodistance Between a Point and a Gyroline 127

2.22 The Gyroangle Bisector Theorem 133

2.23 Möbius Addition and Möbius Gyrogroups 135

2.24 Möbius Gyration 136

2.25 Möbius Gyrovector Spaces 138

2.26 Möbius Points, Gyrolines and Gyrodistance 139

2.27 Linking Möbius Addition to Hyperbolic Geometry 142

2.28 Möbius Gyrovectors, Gyroangles and Gyrotriangles 143

2.29 Gyrovector Space Isomorphism 148

2.30 Möbius Gyrotrigonometry 153

2.31 Exercises 155

3 The Interplay of Einstein Addition and Vector Addition 157

3.1 Extension of Rⁿ_s into Tⁿ⁺¹_s 157

3.2 Scalar Multiplication and Addition in Tⁿ⁺¹_s 162

3.3 Inner Product and Norm in Tⁿ⁺¹_s 163

3.4 Unit Elements of Tⁿ⁺¹ 165

3.5 From Tⁿ⁺¹_s back to Rⁿ_s 173

4 Hyperbolic Barycentric Coordinates and Hyperbolic Triangle Centers 179

4.1 Gyrobarycentric Coordinates in Einstein Gyrovector Spaces 179

4.2 Analogies with Relativistic Mechanics 183

4.3 Gyrobarycentric Coordinates in Möbius Gyrovector Spaces 184

4.4 Einstein Gyromidpoint 187

4.5 Möbius Gyromidpoint 189

4.6 Einstein Gyrotriangle Gyrocentroid 190

4.7 Einstein Gyrotetrahedron Gyrocentroid 197

4.8 Möbius Gyrotriangle Gyrocentroid 199

4.9 Möbius Gyrotetrahedron Gyrocentroid 200

4.10 Foot of a Gyrotriangle Gyroaltitude 201

4.11 Einstein Point to Gyroline Gyrodistance 205

4.12 Möbius Point to Gyroline Gyrodistance 207

4.13 Einstein Gyrotriangle Orthogyrocenter 209

4.14 Möbius Gyrotriangle Orthogyrocenter 219

4.15 Foot of a Gyrotriangle Gyroangle Bisector 224

4.16 Einstein Gyrotriangle Ingyrocenter 229

4.17 Ingyrocenter to Gyrotriangle Side Gyrodistance 237

4.18 Möbius Gyrotriangle Ingyrocenter 240

4.19 Einstein Gyrotriangle Circumgyrocenter 244

4.20 Einstein Gyrotriangle Circumgyroradius 249

4.21 Möbius Gyrotriangle Circumgyrocenter 250

4.22 Comparative Study of Gyrotriangle Gyrocenters 253

4.23 Exercises 257

5 Hyperbolic Incircles and Excircles 259

5.1 Einstein Gyrotriangle Ingyrocenter and Exgyrocenters 259

5.2 Einstein Ingyrocircle and Exgyrocircle Tangency Points 265

5.3 Useful Gyrotriangle Gyrotrigonometric Relations 268

5.4 The Tangency Points Expressed Gyrotrigonometrically 269

5.5 Möbius Gyrotriangle Ingyrocenter and Exgyrocenters 275

5.6 From Gyrotriangle Tangency Points to Gyrotriangle Gyrocenters 280

5.7 Exercises 283

6 Hyperbolic Tetrahedra 285

6.1 Gyrotetrahedron Gyroaltitude 285

6.2 Point Gyroplane Relations 294

6.3 Gyrotetrahedron Ingyrocenter and Exgyrocenters 296

6.4 In-Exgyrosphere Tangency Points 305

6.5 Gyrotrigonometric Gyrobarycentric Coordinates for the Gyrotetrahedron In-Exgyrocenters 307

6.6 Gyrotetrahedron Circumgyrocenter 316

6.7 Exercises 320

7 Comparative Patterns 323

7.1 Gyromidpoints and Gyrocentroids 323

7.2 Two and Three Dimensional Ingyrocenters 326

7.3 Two and Three Dimensional Circumgyrocenters 328

7.4 Tetrahedron Incenter and Excenters 329

7.5 Comparative study of the Pythagorean Theorem 331

7.6 Hyperbolic Heron's Formula 333

7.7 Exercises 334

Notation And Special Symbols 335

Bibliography 337

Index 341