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. GyroangleAngle 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. InExgyroradii Relations  8.6. InExradii Relations  8.7. InExgyrocenter Gyrotrigonometric Gyrobarycentric Representations  8.8. InExcenter 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. ExgyrocenterPointofTangency 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 upperlevel 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. Mathematicalphysicists 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 
