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A gyrovector space approach to hyperbolic geometry

The mere mention of hyperbolic geometry is enough to strike fear in the heart of the undergraduate mathematics and physics student. Some regard themselves as excluded from the profound insights of hyperbolic geometry so that this enormous portion of human achievement is a closed door to them. The mi...

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Bibliographic Details
Main Author: Ungar, Abraham A.
Format: Electronic
Language:English
Published: San Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA) : Morgan & Claypool Publishers, c2009.
Series:Synthesis lectures on mathematics and statistics (Online) ; # 4.
Subjects:
Geometry, Hyperbolic.
Online Access:View fulltext via EzAccess
Table of Contents:
  • Gyrogroups
  • From möbius to gyrogroups
  • Groupoids, loops, groups, and gyrogroups
  • Möbius gyrogroups: from the disc to the ball
  • First gyrogroup theorems
  • The two basic equations of gyrogroups
  • The basic cancellation laws of gyrogroups
  • Commuting automorphisms with gyroautomorphisms
  • The gyrosemidirect product
  • Basic gyration properties
  • An advanced gyrogroup equation
  • Exercises
  • Gyrocommutative gyrogroups
  • Gyrocommutative gyrogroups
  • Möbius gyrogroups
  • Einstein gyrogroups
  • Gyrogroup isomorphism
  • Exercises
  • Gyrovector spaces
  • Definition and first gyrovector space theorems
  • Gyrolines
  • Gyromidpoints
  • Analogies between gyromidpoints and midpoints
  • Gyrogeodesics
  • Möbius gyrovector spaces
  • Möbius gyrolines
  • Einstein gyrovector spaces
  • Einstein gyrolines
  • Einstein gyromidpoints and gyrotriangle gyrocentroids
  • Möbius gyrotriangle gyromedians and gyrocentroids
  • The gyroparallelogram
  • Points, vectors, and gyrovectors
  • The gyroparallelogram addition law of gyrovectors
  • Gyrovector gyrotranslation
  • Gyrovector gyrotranslation composition
  • Gyrovector gyrotranslation and the gyroparallelogram law
  • The möbius gyrotriangle gyroangles
  • Exercises
  • Gyrotrigonometry
  • The gyroangle
  • The gyrotriangle
  • The gyrotriangle addition law
  • Cogyrolines, cogyrotriangles, and cogyroangles
  • The law of gyrocosines
  • The SSS to AAA conversion law
  • Inequalities for gyrotriangles
  • The AAA to SSS conversion law
  • The law of gyrosines
  • The ASA to SAS conversion law
  • The gyrotriangle defect
  • The right gyrotriangle
  • Gyrotrigonometry
  • Gyrodistance between a point and a gyroline
  • The gyrotriangle gyroaltitude
  • The gyrotriangle gyroarea
  • Gyrotriangle similarity
  • The gyroangle bisector theorem
  • The hyperbolic Steiner-Lehmus theorem
  • The Urquhart theorem
  • The hyperbolic Urquhart theorem
  • The gyroparallelogram gyroangles
  • Relativistic mechanical interpretation
  • Gyro-analogies that may reveal the origin of dark matter
  • Newtonian systems of particles
  • Einsteinian systems of particles
  • The relativistic invariant mass paradox
  • Exercises.

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