Analytic hyperbolic geometry : mathematical foundations and applications / Abraham A. Ungar.
This is the first book on analytic hyperbolic geometry, fully analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics just as analytic Euclidean geometry regulates classical mechanics. The book presents a novel gyrovector space approach to analytic hyp...
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| Online Access: |
Full Text (via ProQuest) |
|---|---|
| Main Author: | |
| Format: | eBook |
| Language: | English |
| Published: |
New Jersey :
World Scientific,
©2005.
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| Subjects: |
MARC
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| 100 | 1 | |a Ungar, Abraham A. | |
| 245 | 1 | 0 | |a Analytic hyperbolic geometry : |b mathematical foundations and applications / |c Abraham A. Ungar. |
| 260 | |a New Jersey : |b World Scientific, |c ©2005. | ||
| 300 | |a 1 online resource (xvii, 463 pages) : |b illustrations. | ||
| 336 | |a text |b txt |2 rdacontent. | ||
| 337 | |a computer |b c |2 rdamedia. | ||
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| 504 | |a Includes bibliographical references (pages 445-456) and index. | ||
| 505 | 0 | |a Introduction -- Gyrogroups -- Gyrocommutative gyrogroups -- Gyrogroup extension -- Gyrovectors and cogyrovectors -- Gyrovector spaces -- Rudiments of differential geometry -- Gyrotrigonometry -- Bloch gyrovector of quantum computation -- Special theory of relativity: the analytic hyperbolic geometric viewpoint. | |
| 520 | |a This is the first book on analytic hyperbolic geometry, fully analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics just as analytic Euclidean geometry regulates classical mechanics. The book presents a novel gyrovector space approach to analytic hyperbolic geometry, fully analogous to the well-known vector space approach to Euclidean geometry. A gyrovector is a hyperbolic vector. In the resulting "gyrolanguage" of the book, one attaches the prefix "gyro" to a classical term to mean the analogous term in hyperbolic geometry. The book begins with the definition of gyrogroups, which is fully analogous to the definition of groups. Gyrogroups, both gyrocommutative and nongyrocommutative, abound in group theory. Surprisingly, the seemingly structureless Einstein velocity addition of special relativity turns out to be a gyrocommutative gyrogroup operation. Introducing scalar multiplication, some gyrocommutative gyrogroups of gyrovectors become gyrovector spaces. The latter, in turn, form the setting for analytic hyperbolic geometry just as vector spaces form the setting for analytic Euclidean geometry. By hybrid techniques of differential geometry and gyrovector spaces, it is shown that Einstein (Mobius) gyrovector spaces form the setting for Beltrami-Klein (Poincare) ball models of hyperbolic geometry. Finally, novel applications of Mobius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity, are presented. | ||
| 588 | 0 | |a Print version record. | |
| 650 | 0 | |a Geometry, Hyperbolic |v Textbooks. | |
| 650 | 0 | |a Vector algebra |v Textbooks. | |
| 650 | 7 | |a Geometry, Hyperbolic. |2 fast |0 (OCoLC)fst00940922. | |
| 650 | 7 | |a Vector algebra. |2 fast |0 (OCoLC)fst01164648. | |
| 655 | 7 | |a Textbooks. |2 fast |0 (OCoLC)fst01423863. | |
| 776 | 0 | 8 | |i Print version: |a Ungar, Abraham A. |t Analytic hyperbolic geometry. |d New Jersey : World Scientific, ©2005 |w (DLC) 2005042422. |
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