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Introduction to Hyperbolic Geometry

Author: Arlan Ramsay; Robert D Richtmyer
Publisher: New York, NY : Springer New York, 1995.
Series: Universitext.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Database:WorldCat
Summary:
This text for advanced undergraduates emphasizes the logical connections of the subject. The derivations of formulas from the axioms do not make use of models of the hyperbolic plane until the axioms are shown to be categorical; the differential geometry of surfaces is developed far enough to establish its connections to the hyperbolic plane; and the axioms and proofs use the properties of the real number system to  Read more...
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Details

Genre/Form: Electronic books
Additional Physical Format: Print version:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Arlan Ramsay; Robert D Richtmyer
ISBN: 9781475755855 1475755856
OCLC Number: 851760008
Description: 1 online resource (xii, 289 pages).
Contents: 1 Axioms for Plane Geometry --
2 Some Neutral Theorems of Plane Geometry --
3 Qualitative Description of the Hyperbolic Plane --
4?3 and Euclidean Approximations in?2 --
5 Differential Geometry of Surfaces --
6 Quantitative Considerations --
7 Consistency and Categoricalness of the Hyperbolic Axioms; The Classical Models --
8 Matrix Representation of the Isometry Group --
9 Differential and Hyperbolic Geometry in More Dimensions --
10 Connections with the Lorentz Group of Special Relativity --
11 Constructions by Straightedge and Compass in the Hyperbolic Plane.
Series Title: Universitext.
Responsibility: by Arlan Ramsay, Robert D. Richtmyer.

Abstract:

This text for advanced undergraduates emphasizes the logical connections of the subject. The derivations of formulas from the axioms do not make use of models of the hyperbolic plane until the axioms are shown to be categorical; the differential geometry of surfaces is developed far enough to establish its connections to the hyperbolic plane; and the axioms and proofs use the properties of the real number system to avoid the tedium of a completely synthetic approach. The development includes properties of the isometry group of the hyperbolic plane, tilings, and applications to special relativity. Elementary techniques from complex analysis, matrix theory, and group theory are used, and some mathematical sophistication on the part of students is thus required, but a formal course in these topics is not a prerequisite.
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