1st Edition
Analytic Hyperbolic Geometry in N Dimensions An Introduction
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.
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
Biography
Abraham Albert Ungar
"Anyone who is concerned with hyperbolic geometry should use this wonderful and comprehensive book as a helpful compendium."
—Zentralblatt MATH 1312
