Barycentric Calculus In Euclidean And Hyperbolic Geometry: A Comparative Introduction A Comparative Introduction

Annotation 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 Preface......Page 12 1. Euclidean Barycentric Coordinates and the Classic Triangle Centers......Page 17 1.1 Points, Lines, Distance and Isometries......Page 18 1.2 Vectors, Angles and Triangles......Page 21 1.3 Euclidean Barycentric Coordinates......Page 24 1.4 Analogies with Classical Mechanics......Page 27 1.5 Barycentric Representations are Covariant......Page 28 1.6 Vector Barycentric Representation......Page 30 1.7 Triangle Centroid......Page 33 1.8 Triangle Altitude......Page 35 1.9 Triangle Orthocenter......Page 40 1.10 Triangle Incenter......Page 43 1.11 Triangle Inradius......Page 49 1.12 Triangle Circumcenter......Page 52 1.13 Circumradius......Page 56 1.14 Triangle Incircle and Excircles......Page 58 1.15 Excircle Tangency Points......Page 63 1.16 From Triangle Tangency Points to Triangle Centers......Page 68 1.17 Triangle In-Exradii......Page 71 1.18 A Step Toward the Comparative Study......Page 73 1.19 Tetrahedron Altitude......Page 74 1.20 Tetrahedron Altitude Length......Page 78 1.21 Exercises......Page 79 2. Gyrovector Spaces and Cartesian Models of Hyperbolic Geometry......Page 81 2.1 Einstein Addition......Page 82 2.2 Einstein Gyration......Page 86 2.3 From Einstein Velocity Addition to Gyrogroups......Page 89 2.4 First Gyrogroup Theorems......Page 93 2.5 The Two Basic Equations of Gyrogroups......Page 98 2.6 Einstein Gyrovector Spaces......Page 102 2.7 Gyrovector Spaces......Page 105 2.8 Einstein Points, Gyrolines and Gyrodistance......Page 111 2.9 Linking Einstein Addition to Hyperbolic Geometry......Page 115 2.10 Einstein Gyrovectors, Gyroangles and Gyrotriangles......Page 117 2.11 The Law of Gyrocosines......Page 122 2.12 The SSS to AAA Conversion Law......Page 124 2.13 Inequalities for Gyrotriangles......Page 125 2.14 The AAA to SSS Conversion Law......Page 127 2.16 The ASA to SAS Conversion Law......Page 131 2.17 Gyrotriangle Defect......Page 132 2.18 Right Gyrotriangles......Page 134 2.19 Einstein Gyrotrigonometry and Gyroarea......Page 136 2.20 Gyrotriangle Gyroarea Addition Law......Page 140 2.21 Gyrodistance Between a Point and a Gyroline......Page 143 2.22 The Gyroangle Bisector Theorem......Page 149 2.23 Mobius Addition and Mobius Gyrogroups......Page 151 2.24 Mobius Gyration......Page 152 2.25 Mobius Gyrovector Spaces......Page 154 2.26 Mobius Points, Gyrolines and Gyrodistance......Page 155 2.27 Linking Mobius Addition to Hyperbolic Geometry......Page 158 2.28 Mobius Gyrovectors, Gyroangles and Gyrotriangles......Page 159 2.29 Gyrovector Space Isomorphism......Page 164 2.30 Mobius Gyrotrigonometry......Page 169 2.31 Exercises......Page 171 3.1 Extension of R into Tn+1......Page 173 3.2 Scalar Multiplication and Addition in Tn+1......Page 178 3.3 Inner Product and Norm in Tn+1......Page 179 3.4 Unit Elements of Tn+1......Page 181 3.5 From Tn+1 back to R......Page 189 4.1 Gyrobarycentric Coordinates in Einstein Gyrovector Spaces......Page 195 4.2 Analogies with Relativistic Mechanics......Page 199 4.3 Gyrobarycentric Coordinates in Mobius Gyrovector Spaces......Page 200 4.4 Einstein Gyromidpoint......Page 203 4.5 Mobius Gyromidpoint......Page 205 4.6 Einstein Gyrotriangle Gyrocentroid......Page 206 4.7 Einstein Gyrotetrahedron Gyrocentroid......Page 213 4.8 Mobius Gyrotriangle Gyrocentroid......Page 215 4.9 Mobius Gyrotetrahedron Gyrocentroid......Page 216 4.10 Foot of a Gyrotriangle Gyroaltitude......Page 217 4.11 Einstein Point to Gyroline Gyrodistance......Page 221 4.12 Mobius Point to Gyroline Gyrodistance......Page 223 4.13 Einstein Gyrotriangle Orthogyrocenter......Page 225 4.14 Mobius Gyrotriangle Orthogyrocenter......Page 235 4.15 Foot of a Gyrotriangle Gyroangle Bisector......Page 240 4.16 Einstein Gyrotriangle Ingyrocenter......Page 245 4.17 Ingyrocenter to Gyrotriangle Side Gyrodistance......Page 253 4.18 Mobius Gyrotriangle Ingyrocenter......Page 256 4.19 Einstein Gyrotriangle Circumgyrocenter......Page 260 4.20 Einstein Gyrotriangle Circumgyroradius......Page 265 4.21 Mobius Gyrotriangle Circumgyrocenter......Page 266 4.22 Comparative Study of Gyrotriangle Gyrocenters......Page 269 4.23 Exercises......Page 273 5.1 Einstein Gyrotriangle Ingyrocenter and Exgyrocenters......Page 275 5.2 Einstein Ingyrocircle and Exgyrocircle Tangency Points......Page 281 5.3 Useful Gyrotriangle Gyrotrigonometric Relations......Page 284 5.4 The Tangency Points Expressed Gyrotrigonometrically......Page 285 5.5 M obius Gyrotriangle Ingyrocenter and Exgyrocenters......Page 291 5.6 From Gyrotriangle Tangency Points to Gyrotriangle Gyrocenters......Page 296 5.7 Exercises......Page 299 6.1 Gyrotetrahedron Gyroaltitude......Page 301 6.2 Point Gyroplane Relations......Page 310 6.3 Gyrotetrahedron Ingyrocenter and Exgyrocenters......Page 312 6.4 In-Exgyrosphere Tangency Points......Page 321 6.5 Gyrotrigonometric Gyrobarycentric Coordinates for the Gyrotetrahedron In-Exgyrocenters......Page 323 6.6 Gyrotetrahedron Circumgyrocenter......Page 332 6.7 Exercises......Page 336 7.1 Gyromidpoints and Gyrocentroids......Page 339 7.2 Two and Three Dimensional Ingyrocenters......Page 342 7.3 Two and Three Dimensional Circumgyrocenters......Page 344 7.4 Tetrahedron Incenter and Excenters......Page 345 7.5 Comparative study of the Pythagorean Theorem......Page 347 7.6 Hyperbolic Heron's Formula......Page 349 7.7 Exercises......Page 350 Notation And Special Symbols......Page 351 Bibliography......Page 353 Index......Page 357 1. Euclidean barycentric coordinates and the classic triangle centers. 1.1. Points, lines, distance and isometries. 1.2. Vectors, angles and triangles. 1.3. Euclidean barycentric coordinates. 1.4. Analogies with classical mechanics. 1.5. Barycentric representations are covariant. 1.6. Vector barycentric representation. 1.7. Triangle centroid. 1.8. Triangle altitude. 1.9. Triangle orthocenter. 1.10. Triangle incenter. 1.11. Triangle inradius. 1.12. Triangle circumcenter. 1.13. Circumradius. 1.14. Triangle incircle and excircles. 1.15. Excircle tangency points. 1.16. From triangle tangency points to triangle centers. 1.17. Triangle in-exradii. 1.18. A step toward the comparative study. 1.19. Tetrahedron altitude. 1.20. Tetrahedron altitude length. 1.21. Exercises -- 2. Gyrovector spaces and Cartesian models of hyperbolic geometry. 2.1. Einstein addition. 2.2. Einstein gyration. 2.3. From Einstein velocity addition to gyrogroups. 2.4. First gyrogroup theorems. 2.5. The two basic equations of gyrogroups. 2.6. Einstein gyrovector spaces. 2.7. Gyrovector spaces. 2.8. Einstein points, gyrolines and gyrodistance. 2.9. Linking Einstein addition to hyperbolic geometry. 2.10. Einstein gyrovectors, gyroangles and gyrotriangles. 2.11. The law of gyrocosines. 2.12. The SSS to AAA conversion law. 2.13. Inequalities for gyrotriangles. 2.14. The AAA to SSS conversion law. 2.15. The law of gyrosines. 2.16. The ASA to SAS conversion law. 2.17. Gyrotriangle defect. 2.18. Right gyrotriangles. 2.19. Einstein gyrotrigonometry and gyroarea. 2.20. Gyrotriangle gyroarea addition law. 2.21. Gyrodistance between a point and a gyroline. 2.22. The gyroangle bisector theorem. 2.23. Mobius addition and Mobius gyrogroups. 2.24. Mobius gyration. 2.25. Mobius gyrovector spaces. 2.26. Mobius points, gyrolines and gyrodistance. 2.27. Linking Mobius addition to hyperbolic geometry. 2.28. Mobius gyrovectors, gyroangles and gyrotriangles. 2.29. Gyrovector space isomorphism. 2.30. Mobius gyrotrigonometry. 2.31. Exercises -- 3. The interplay of Einstein addition and vector addition. 3.1. Extension of R[symbol] into T[symbol]. 3.2. Scalar multiplication and addition in T[symbol]. 3.3. Inner product and norm in T[symbol]. 3.4. Unit elements of T[symbol]. 3.5. From T[symbol] back to R[symbol] -- 4. Hyperbolic Barycentric coordinates and hyperbolic triangle centers. 4.1. Gyrobarycentric coordinates in Einstein gyrovector spaces. 4.2. Analogies with relativistic mechanics. 4.3. Gyrobarycentric coordinates in Mobius gyrovector spaces. 4.4. Einstein gyromidpoint. 4.5. Mobius gyromidpoint. 4.6. Einstein gyrotriangle gyrocentroid. 4.7. Einstein gyrotetrahedron gyrocentroid. 4.8. Mobius gyrotriangle gyrocentroid. 4.9. Mobius gyrotetrahedron gyrocentroid. 4.10. Foot of a gyrotriangle gyroaltitude. 4.11. Einstein point to gyroline gyrodistance. 4.12. Mobius point to gyroline gyrodistance. 4.13. Einstein gyrotriangle orthogyrocenter. 4.14. Mobius gyrotriangle orthogyrocenter. 4.15. Foot of a gyrotriangle gyroangle bisector. 4.16. Einstein gyrotriangle ingyrocenter. 4.17. Ingyrocenter to gyrotriangle side gyrodistance. 4.18. Mobius gyrotriangle ingyrocenter. 4.19. Einstein gyrotriangle circumgyrocenter. 4.20. Einstein gyrotriangle circumgyroradius. 4.21. Mobius gyrotriangle circumgyrocenter. 4.22. Comparative study of gyrotriangle gyrocenters. 4.23. Exercises -- 5. Hyperbolic incircles and excircles. 5.1. Einstein gyrotriangle ingyrocenter and exgyrocenters. 5.2. Einstein ingyrocircle and exgyrocircle tangency points. 5.3. Useful gyrotriangle gyrotrigonometric relations. 5.4. The tangency points expressed gyrotrigonometrically. 5.5. Mobius gyrotriangle ingyrocenter and exgyrocenters. 5.6. From gyrotriangle tangency points to gyrotriangle gyrocenters. 5.7. Exercises -- 6. Hyperbolic tetrahedra. 6.1. Gyrotetrahedron gyroaltitude. 6.2. Point gyroplane relations. 6.3. Gyrotetrahedron ingyrocenter and exgyrocenters. 6.4. In-exgyrosphere tangency points. 6.5. Gyrotrigonometric gyrobarycentric coordinates for the gyrotetrahedron in-exgyrocenters. 6.6. Gyrotetrahedron circumgyrocenter. 6.7. Exercises -- 7. Comparative patterns. 7.1. Gyromidpoints and gyrocentroids. 7.2. Two and three dimensional ingyrocenters. 7.3. Two and three dimensional circumgyrocenters. 7.4. Tetrahedron incenter and excenters. 7.5. Comparative study of the Pythagorean Theorem. 7.6. Hyperbolic Heron's formula. 7.7. Exercises