Barycentric Calculus In Euclidean And Hyperbolic Geometry: A Comparative Introduction

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Abraham Albert Ungar (North Dakota State Univ, Usa)

Barycentric Calculus In Euclidean And Hyperbolic Geometry

A Comparative Introduction

Abraham Albert Ungar (North Dakota State Univ, Usa)

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Hardback

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English

World Scientific Publishing Co Pte Ltd
26 October 2025

Euclidean geometry; Mathematical physics

Summary
Details

This unique and richly illustrated book explores barycentric calculus, a geometric method grounded in the concept of the center of gravity. Used to elegantly determine triangle centers through weighted points, barycentric coordinates have long revealed deep insights in Euclidean geometry. Now, this book extends those insights to the fascinating realm of hyperbolic geometry, building a powerful bridge between classical and modern mathematical worlds.

In Euclidean geometry, over 3,000 triangle centers have been identified using barycentric coordinates. This book introduces readers to their hyperbolic analogs, uncovering remarkable parallels between triangle centers in Bolyai-Lobachevsky geometry and their Euclidean counterparts. The author's innovative use of Cartesian coordinates, trigonometry, and vector algebra — adapted for hyperbolic geometry — equips readers with familiar yet powerful tools to explore unfamiliar terrain.

At the heart of the book is the development of hyperbolic barycentric coordinates, or gyrobarycentric coordinates, within the framework of gyrovector spaces — a novel algebraic structure emerging from Einstein's velocity addition and Möbius addition. These gyrovectors underpin the Klein and Poincaré ball models of hyperbolic geometry, just as traditional vectors underlie analytic Euclidean geometry.

Whether you are a researcher in geometry, mathematical physics, or relativity, or simply fascinated by the deep structure of space, this book offers a groundbreaking approach to analytic hyperbolic geometry through barycentric and gyrobarycentric coordinates.

By: Abraham Albert Ungar (North Dakota State Univ Usa)
Imprint: World Scientific Publishing Co Pte Ltd
Country of Publication: Singapore
Edition: Second Edition
ISBN: 9789819821297
ISBN 10: 9819821290
Pages: 400
Publication Date: 26 October 2025
Audience: College/higher education , Professional and scholarly , Primary , Undergraduate
Format: Hardback
Publisher's Status: Active