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Author	Ungar, Abraham A.
Title	A gyrovector space approach to hyperbolic geometry / Abraham Albert Ungar.
OCLC	200901MAS004
ISBN	9781598298239 (electronic bk.)
	9781598298222 (pbk.)
ISBN/ISSN	10.2200/S00175ED1V01Y200901MAS004 doi
Publisher	San Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA) : Morgan & Claypool Publishers, [2009]
	�2009
Description	1 electronic text (xii, 182 pages : illustrations\.) : digital file.
LC Subject heading/s	Geometry, Hyperbolic.
SUBJECT	Hyperbolic geometry.
	Analogies between Euclidean and hyperbolic geometry.
	Poincar� ball model of hyperbolic geometry.
	Beltrami-Klein ball model of hyperbolic geometry.
	M�bius transformations of the complex open unit disc.
	M�bius addition, from M�bius to gyrogroups.
	Gyrogroups.
	Gyrocommutative gyrogroups.
	Gyrovectors.
	Gyrovector spaces.
	Gyrotrigonometry.
	Einstein relativistic velocity addition.
	Relativistic stellar aberration.
	Dark matter.
System details note	Mode of access: World Wide Web.
	System requirements: Adobe Acrobat reader.
Bibliography	Includes bibliographical references (pages 173-177) and index.
Contents	Gyrogroups -- From m�bius to gyrogroups -- Groupoids, loops, groups, and gyrogroups -- M�bius 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 -- M�bius gyrogroups -- Einstein gyrogroups -- Gyrogroup isomorphism -- Exercises -- Gyrovector spaces -- Definition and first gyrovector space theorems -- Gyrolines -- Gyromidpoints -- Analogies between gyromidpoints and midpoints -- Gyrogeodesics -- M�bius gyrovector spaces -- M�bius gyrolines -- Einstein gyrovector spaces -- Einstein gyrolines -- Einstein gyromidpoints and gyrotriangle gyrocentroids -- M�bius 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 m�bius 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.
Restrictions	Abstract freely available; full-text restricted to subscribers or individual document purchasers.
	Access may be restricted to authorized users only.
	Unlimited user license access
NOTE	Compendex.
	INSPEC.
	Google book search.
Summary	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 mission of this book is to open that door by making the hyperbolic geometry of Bolyai and Lobachevsky, as well as the special relativity theory of Einstein that it regulates, accessible to a wider audience in terms of novel analogies that the modern and unknown share with the classical and familiar. These novel analogies that this book captures stem from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Remarkably, the mere introduction of Thomas gyration turns Euclidean geometry into hyperbolic geometry, and reveals mystique analogies that the two geometries share. Accordingly, Thomas gyration gives rise to the prefix "gyro" that is extensively used in the gyrolanguage of this book, giving rise to terms like gyrocommutative and gyroassociative binary operations in gyrogroups, and gyrovectors in gyrovector spaces. Of particular importance is the introduction of gyrovectors into hyperbolic geometry, where they are equivalence classes that add according to the gyroparallelogram law in full analogy with vectors, which are equivalence classes that add according to the parallelogram law. A gyroparallelogram, in turn, is a gyroquadrilateral the two gyrodiagonals of which intersect at their gyromidpoints in full analogy with a parallelogram, which is a quadrilateral the two diagonals of which intersect at their midpoints.
NOTE	Google scholar.
Additional physical form available note	Also available in print.
General note	Part of: Synthesis digital library of engineering and computer science.
	Title from PDF t.p. (viewed on January 8, 2009).
	Series from website.

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