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Barycentric Calculus in Euclidean and Hyperbolic Geometry cover

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.

Sample Chapter(s)
Chapter 1: Euclidean Barycentric Coordinates and the Classic Triangle Centers (478 KB)


Contents:
  • Euclidean Barycentric Coordinates and the Classic Triangle Centers
  • Gyrovector Spaces and Cartesian Models of Hyperbolic Geometry
  • The Interplay of Einstein Addition and Vector Addition
  • Hyperbolic Barycentric Coordinates and Hyperbolic Triangle Centers
  • Hyperbolic Incircles and Excircles
  • Hyperbolic Tetrahedra
  • Comparative Patterns

Readership: Undergraduate and graduate students of mathematics who are familiar with the basic vector space approach to Euclidean geometry. Researchers and academics in geometry, algebra, and mathematical physics.
Free Access
FRONT MATTER
  • Pages:i–xiv

https://doi.org/10.1142/9789814304948_fmatter

No Access
Euclidean Barycentric Coordinates and the Classic Triangle Centers
  • Pages:1–64

https://doi.org/10.1142/9789814304948_0001

No Access
Gyrovector Spaces and Cartesian Models of Hyperbolic Geometry
  • Pages:65–156

https://doi.org/10.1142/9789814304948_0002

No Access
The Interplay of Einstein Addition and Vector Addition
  • Pages:157–178

https://doi.org/10.1142/9789814304948_0003

No Access
Hyperbolic Barycentric Coordinates and Hyperbolic Triangle Centers
  • Pages:179–257

https://doi.org/10.1142/9789814304948_0004

No Access
Hyperbolic Incircles and Excircles
  • Pages:259–284

https://doi.org/10.1142/9789814304948_0005

No Access
Hyperbolic Tetrahedra
  • Pages:285–321

https://doi.org/10.1142/9789814304948_0006

No Access
Comparative Patterns
  • Pages:323–334

https://doi.org/10.1142/9789814304948_0007

Free Access
BACK MATTER
  • Pages:335–344

https://doi.org/10.1142/9789814304948_bmatter

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