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Hyperbolic Triangle Centers

The Special Relativistic Approach

  • Book
  • © 2010

Overview

Authors:
  1. A.A. Ungar

    1. Dept. Mathematics, North Dakota State University, Fargo, USA

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  • Continuation of A. Ungar successful work on hyperbolic geometry, now with introduction of hyperbolic barycentric coordinates
  • Proves how, contrary to general belief, Einstein’s relativistic mass meshes up well with Minkowski’s four-vector formalism of special relativity
  • Sets the ground for investigating hyperbolic triangle centers analytically with respect to its hyperbolic triangle vertices
  • Includes supplementary material: sn.pub/extras

Part of the book series: Fundamental Theories of Physics (FTPH, volume 166)

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Table of contents (10 chapters)

  1. Front Matter

    Pages I-XVI
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  2. The Special Relativistic Approach To Hyperbolic Geometry

    1. Front Matter

      Pages 1-1
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    2. Einstein Gyrogroups

      • A. A. Ungar
      Pages 3-43

    3. Einstein Gyrovector Spaces

      • A. A. Ungar
      Pages 45-57

    4. When Einstein Meets Minkowski

      • A. A. Ungar
      Pages 59-81

  3. Mathematical Tools For Hyperbolic Geometry

    1. Front Matter

      Pages 83-83
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    2. Euclidean and Hyperbolic Barycentric Coordinates

      • A. A. Ungar
      Pages 85-115

    3. Gyrovectors

      • A. A. Ungar
      Pages 117-125

    4. Gyrotrigonometry

      • A. A. Ungar
      Pages 127-150

  4. Hyperbolic Triangle Centers

    1. Front Matter

      Pages 151-151
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    2. Gyrotriangle Gyrocenters

      • A. A. Ungar
      Pages 153-217

    3. Gyrotriangle Exgyrocircles

      • A. A. Ungar
      Pages 219-267

    4. Gyrotriangle Gyrocevians

      • A. A. Ungar
      Pages 269-299

    5. Epilogue

      • A. A. Ungar
      Pages 301-308

  5. Back Matter

    Pages 309-319
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Keywords

About this book

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.

Authors and Affiliations

  • Dept. Mathematics, North Dakota State University, Fargo, USA

    A.A. Ungar

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