Hyperbolic Triangle Centers (eBook)

The Special Relativistic Approach

(Autor)

eBook Download: PDF
2010 | 2010
XVI, 319 Seiten
Springer Netherland (Verlag)
978-90-481-8637-2 (ISBN)

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Hyperbolic Triangle Centers - A.A. Ungar
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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 four-vector 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.
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 four-vector 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 velocityspace of cosmology.

The Special Relativistic Approach To Hyperbolic Geometry.- Einstein Gyrogroups.- Einstein Gyrovector Spaces.- When Einstein Meets Minkowski.- Mathematical Tools For Hyperbolic Geometry.- Euclidean and Hyperbolic Barycentric Coordinates.- Gyrovectors.- Gyrotrigonometry.- Hyperbolic Triangle Centers.- Gyrotriangle Gyrocenters.- Gyrotriangle Exgyrocircles.- Gyrotriangle Gyrocevians.- Epilogue.

Erscheint lt. Verlag 18.6.2010
Reihe/Serie Fundamental Theories of Physics
Fundamental Theories of Physics
Zusatzinfo XVI, 319 p.
Verlagsort Dordrecht
Sprache englisch
Themenwelt Mathematik / Informatik Mathematik Angewandte Mathematik
Mathematik / Informatik Mathematik Geometrie / Topologie
Naturwissenschaften Physik / Astronomie Astronomie / Astrophysik
Naturwissenschaften Physik / Astronomie Relativitätstheorie
Naturwissenschaften Physik / Astronomie Theoretische Physik
Technik
Schlagworte Application special relativity • Barycentric coordinates • Examining hyperbolic triangle center • Four-vector • Hyperbolic barycentric coordinates • Hyperbolic coordinates • Hyperbolic Geometry • Hyperbolic tri • Hyperbolic triangle centers • Hyperbolic triangle ve • Hyperbolic triangle vertex • Relativity • RMS • Special relativity • Special theory of relativity • theoretical physics
ISBN-10 90-481-8637-4 / 9048186374
ISBN-13 978-90-481-8637-2 / 9789048186372
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