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Author Ungar, Abraham A.

Title Hyperbolic triangle centers : the special relativistic approach / by Abraham A. Ungar
Published Dordrecht ; London : Springer, [2010]
©2010
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Description 1 online resource (xvi, 319 pages)
Series Fundamental theories of physics ; 166
Fundamental theories of physics ; 166
Contents Note continued: 4.4. Triangle Centroid -- 4.5. Gyromidpoint -- 4.6. Hyperbolic Lever Law Relation -- 4.7. Gyrotriangle Gyrocentroid -- 4.8. Analogies Between Centroids and Gyrocentroids -- 4.9. Gyrodistance in Gyrobarycentric Coordinates -- 4.10. Gyrolines in Gyrobarycentric Coordinates -- 4.11. Problems -- 5. Gyrovectors -- 5.1. Points and Vectors in Euclidean Geometry -- 5.2. Points and Gyrovectors in Hyperbolic Geometry -- 5.3. Einstein Gyroparallelogram -- 5.4. Gyroparallelogram Law -- 6. Gyrotrigonometry -- 6.1. Gyroangles -- 6.2. Gyroangle-Angle Relationship -- 6.3. Law of Gyrocosines -- 6.4. SSS to AAA Conversion Law -- 6.5. Inequalities for Gyrotriangles -- 6.6. AAA to SSS Conversion Law -- 6.7. Law of Gyrosines -- 6.8. ASA to SAS Conversion Law -- 6.9. Gyrotriangle Defect -- 6.10. Right Gyrotriangles -- 6.11. Gyrotrigonometry -- 6.12. Problems -- pt. III Hyperbolic Triangle Centers -- 7. Gyrotriangle Gyrocenters -- 7.1. Gyrotriangle Circumgyrocenter -- 7.2. Triangle Circumcenter -- 7.3. Gyrocircle -- 7.4. Gyrotriangle Circumgyroradius -- 7.5. Gyrocircle Through Three Points -- 7.6. Inscribed Gyroangle Theorem -- 7.7. Gyrotriangle Gyroangle Bisector Foot -- 7.8. Gyrotriangle Ingyrocenter -- 7.9. Gyrotriangle Gyroaltitude Foot -- 7.10. Gyrotriangle Gyroaltitude -- 7.11. Gyrotriangle Ingyroradius -- 7.12. Useful Gyrotriangle Gyrotrigonometric Identities -- 7.13. Gyrotriangle Circumgyrocenter Gyrodistance from Sides -- 7.14. Ingyrocircle Points of Tangency -- 7.15. Unlikely Concurrence -- 7.16. Gergonne Gyropoint -- 7.17. Gyrotriangle Orthogyrocenter -- 7.18. Gyrodistance Between O and I -- 7.19. Problems -- 8. Gyrotriangle Exgyrocircles -- 8.1. Introduction -- 8.2. Gyrotriangle Exgyrocircles and Ingyrocircles -- 8.3. Existence of Gyrotriangle Exgyrocircles
Note continued: 8.4. Exgyroradius and Ingyroradius -- 8.5. In-Exgyroradii Relations -- 8.6. In-Exradii Relations -- 8.7. In-Exgyrocenter Gyrotrigonometric Gyrobarycentric Representations -- 8.8. In-Excenter Trigonometric Barycentric Representations -- 8.9. Exgyrocircle Points of Tangency, Part I -- 8.10. Excircle Points of Tangency, Part I -- 8.11. Left Gyrotranslated Exgyrocircles -- 8.12. Nagel Gyropoint -- 8.13. Exgyrocircle Points of Tangency, Part II -- 8.14. Excircle Points of Tangency, Part II -- 8.15. Gyrodistance Between Gyrotriangle Tangency Points -- 8.15.1. Gyrodistance Between T12 and T13 -- 8.15.2. Gyrodistance Between T1 and T12, T13 -- 8.15.3. Resulting Gyrodistances Between Tangency Points -- 8.16. Exgyrocircle Gyroangles -- 8.17. Exgyrocircle Gyroangle Sum -- 8.18. Exgyrocenter-Point-of-Tangency Gyrocenter -- 8.19. Problems -- 9. Gyrotriangle Gyrocevians -- 9.1. Gyrocevians and the Hyperbolic Theorem of Ceva -- 9.2. Gyrocevian Gyroangles Theorem -- 9.3. Gyrocevian Gyrolength -- 9.4. Cevian Length -- 9.5. Special Gyrocevian -- 9.6. Brocard Gyropoints -- 9.7. Gyrocevian Concurrency Condition -- 9.8. Problems -- 10. Epilogue -- 10.1. Introduction -- 10.2. Stellar Aberration -- 10.3. On the Future of Special Relativity and Hyperbolic Geometry
Summary After A. Ungar had introduced vector algebra and Cartesian coordinates into hyperbolic geometry in his earlier books, along with novel applications in Einstein's special theory of relativity, the purpose of his new book is to introduce hyperbolic barycentric coordinates, another important concept to embed Euclidean geometry into hyperbolic geometry. It will be demonstrated that, in full analogy to classical mechanics where barycentric coordinates are related to the Newtonian mass, barycentric coordinates are related to the Einsteinian relativistic mass in hyperbolic geometry. Contrary to general belief, Einstein's relativistic mass hence meshes up extraordinarily well with Minkowski's fourvector formalism of special relativity
In Euclidean geometry, barycentric coordinates can be used to determine various triangle centers. While there are many known Euclidean triangle centers, only few hyperbolic triangle centers are known, and none of the known hyperbolic triangle centers has been determined analytically with respect to its hyperbolic triangle vertices. In his recent research, the author set the ground for investigating hyperbolic triangle centers via hyperbolic barycentric coordinates, and one of the purposes of this book is to initiate a study of hyperbolic triangle centers in full analogy with the rich study of Euclidean triangle centers. Owing to its novelty, the book is aimed at a large audience: it can be enjoyed equally by upper-level undergraduates, graduate students, researchers and academics in geometry, abstract algebra, theoretical physics and astronomy. For a fruitful reading of this book, familiarity with Euclidean geometry is assumed. Mathematical-physicists and theoretical physicists are likely to enjoy the study of Einstein's special relativity in terms of its underlying hyperbolic geometry. Geometers may enjoy the hunt for new hyperbolic triangle centers and, finally, astronomers may use hyperbolic barycentric coordinates in the velocity space of cosmology. --Book Jacket
Bibliography Includes bibliographical references and indexes
Notes Print version record
Subject Geometry, Hyperbolic.
Special relativity (Physics)
Form Electronic book
LC no. 2010930171
ISBN 9789048186372
9048186374