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Analytic Hyperbolic Geometry in N Dimensions

An Introduction

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## Book Description

The concept of the Euclidean simplex is important in the study of n-dimensional Euclidean geometry. This book introduces for the first time the concept of hyperbolic simplex as an important concept in n-dimensional hyperbolic geometry.

Following the emergence of his gyroalgebra in 1988, the author crafted gyrolanguage, the algebraic language that sheds natural light on hyperbolic geometry and special relativity. Several authors have successfully employed the author’s gyroalgebra in their exploration for novel results. Françoise Chatelin noted in her book, and elsewhere, that the computation language of Einstein described in this book plays a universal computational role, which extends far beyond the domain of special relativity.

This book will encourage researchers to use the author’s novel techniques to formulate their own results. The book provides new mathematical tools, such as hyperbolic simplexes, for the study of hyperbolic geometry in n dimensions. It also presents a new look at Einstein’s special relativity theory.

## Table of Contents

List of Figures

Preface

Author’s Biography

Introduction

Gyrovector Spaces in the Service of Abalytic Hyperbolic Geometry

When Two Counterintuitive Theories Meet

The Fascinating Rich Mathematical Life of Einstein’s Velocity Addition Law

Parts of the Book

**Einstein Gyrogroups and Gyrovector Spaces**

Einstein Gyrogroups

Introduction

Einstein Velocity Addition

Einstein Addition for Computer Algebra

Thomas Precession Angle

Einstein Addition With Respect to Cartesian Coordinates

Einstein Addition Vs. Vector Addition

Gyrations

Gyration Angles

From Einstein Velocity Addition to Gyrogroups

Gyrogroup Cooperation (Coaddition)

First Gyrogroup Properties

Elements of Gyrogroup Theory

The Two Basic Gyrogroup Equations

The Basic Gyrogroup Cancellation Laws

Automorphisms and Gyroautomorphisms

Gyrosemidirect Product

Basic Gyration Properties

An Advanced Gyrogroup Equation

Gyrocommutative Gyrogroups

Problems

Einstein Gyrovector Spaces 65

The Abstract Gyrovector Space

Einstein Gyrovector Spaces

Einstein Addition and Differential Geometry

Euclidean Lines

Gyrolines – The Hyperbolic Lines

Gyroangles – The Hyperbolic Angles

Euclidean Isometries

The Group of Euclidean Motions

Gyroisometries – The Hyperbolic Isometries

Gyromotions – The Motions of Hyperbolic Geometry

Problems

Relativistic Mass Meets Hyperbolic Geometry

Lorentz Transformation and Einstein Addition

Mass of Particle Systems

Resultant Relativistically Invariant Mass

Problems

**Mathematical Tools for Hyperbolic Geometry**

Barycentric and Gyrobarycentric Coordinates

Barycentric Coordinates

Segments

Gyrobarycentric Coordinates

Uniqueness of Gyrobarycentric Representations

Gyrovector Gyroconvex Span

Gyrosegments

Triangle Centroid

Gyromidpoint

Gyroline Boundary points

Gyrotriangle Gyrocentroid

Gyrodistance in Gyrobarycentric Coordinates

Gyrolines in Gyrobarycentric Coordinates

Problems

Gyroparallelograms and Gyroparallelotopes

The Parallelogram Law

Einstein Gyroparallelograms

The Gyroparallelogram Law

The Higher-Dimensional Gyroparallelotope Law

Gyroparallelotopes

Gyroparallelotope Gyrocentroid

Gyroparallelotope Formal Definition and Theorem

Low Dimensional Gyroparallelotopes

Hyperbolic Plane Separation

GPSA for the Einstein Gyroplane

Problems

Gyrotrigonometry

Gyroangles

Gyroangle – Angle Relationship

The Law of Gyrocosines

The SSS to AAA Conversion Law

Inequalities for Gyrotriangles

The AAA to SSS Conversion Law

The Law of Sines/Gyrosines

The Law of Gyrosines

The ASA to SAS Conversion Law

Gyrotriangle Defect

Right Gyrotriangles

Gyrotrigonometry

Gyroangle of Parallelism

Useful Gyrotriangle Gyrotrigonometric Identities

A Determinantal Pattern

Problems

**Hyperbolic Triangles and Circles**

Gyrotriangles and Gyrocircles

Gyrocircles

Gyrotriangle Circumgyrocenter

Triangle Circumcenter

Gyrotriangle Circumgyroradius

Triangle Circumradius

The Gyrocircle Through Three Points

The Inscribed Gyroangle Theorem I

The Inscribed Gyroangle Theorem II

Gyrocircle Gyrotangent Gyrolines

Semi-Gyrocircle Gyrotriangles

Problems

Gyrocircle Theorems

The Gyrotangent–Gyrosecant Theorem

The Intersecting Gyrosecants Theorem

Gyrocircle Gyrobarycentric Representation

Gyrocircle Interior and Exterior Points

Circle Barycentric Representation

Gyrocircle Gyroline Intersection

Gyrocircle–Gyroline Tangency Points

Gyrocircle Gyrotangent Gyrolength

Circle–Line Tangency Points

Circumgyrocevians

Gyrodistances Related to the Gyrocevian

A Gyrodistance Related to the Circumgyrocevian

Circumgyrocevian Gyrolength

The Intersecting Gyrochords Theorem

Problems

**Hyperbolic Simplices, Hyperplanes and Hyperspheres in N**

Dimensions

Gyrosimplices

Gyrotetrahedron Circumgyrocenter

Gyrotetrahedron Circumgyroradius

Gyrosimplex Gyrocentroid

Gamma Matrices

Gyrosimplex Gyroaltitudes

Gyrosimplex Circumhypergyrosphere

The Gyrosimplex Constant

Point to Gyrosimplex Gyrodistance

Cramer’s Rule

Point to Gyrosimplex Perpendicular Projection

Gyrosimplex In-Exgyrocenters and In-Exgyroradii

Gyrotriangle In-Exgyrocenters

Gyrosimplex Gyrosymmedian

Problems

Gyrosimplex Gyrovolume

Gyrovolume

Problems

**Hyperbolic Ellipses and Hyperbolas **

Gyroellipses and Gyrohyperbolas

Gyroellipses – A Gyrobarycentric Representation

Gyroellipses – Gyrotrigonometric Gyrobarycentric Representation

Gyroellipse Major Vertices

Gyroellipse Minor Vertices

Canonical Gyroellipses

Gyrobarycentric Representation of Canonical Gyroellipses

Barycentric Representation of Canonical Ellipses

Some Properties of Canonical Gyroellipses

Canonical Gyroellipses and Ellipses

Canonical Gyroellipse Equation

A Gyrotrigonometric Constant of the Gyroellipse

Ellipse Eccentricity

Gyroellipse Gyroeccentricity

Gyrohyperbolas – A Gyrobarycentric Representation

Problems

**Thomas Precession**

Thomas Precession

Introduction

The Gyrotriangle Defect and Thomas Precession

Thomas Precession

Thomas Precession Matrix

Thomas Precession Graphical Presentation

Thomas Precession Angle

Thomas Precession Frequency

Thomas Precession and Boost Composition

Thomas Precession Angle and Generating Angle have Opposite Signs

Problems

Bibliography

Index

## Reviews

"Anyone who is concerned with hyperbolic geometry should use this wonderful and comprehensive book as a helpful compendium."

—Zentralblatt MATH1312