ANALYTIC HYPERBOLIC GEOMETRY: MATHEMATICAL FOUNDATIONS AND APPLICATIONS - Hardcover
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. Gyrovectors are equivalence classes of directed gyrosegments that add according to the gyroparallelogram law just as vectors are equivalence classes of directed segments that add according to the parallelogram law. 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 prefix stems from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Gyrolanguage turns out to be the language one needs to articulate novel analogies that the classical and the modern in this book share.The scope of analytic hyperbolic geometry that the book presents is cross-disciplinary, involving nonassociative algebra, geometry and physics. As such, it is naturally compatible with the special theory of relativity and, particularly, with the nonassociativity of Einstein velocity addition law. Along with analogies with classical results that the book emphasizes, there are remarkable disanalogies as well. Thus, for instance, unlike Euclidean triangles, the sides of a hyperbolic triangle are uniquely determined by its hyperbolic angles. Elegant formulas for calculating the hyperbolic side-lengths of a hyperbolic triangle in terms of its hyperbolic angles are presented in the book.The book begins with the definition of gyrogroups, which is fully analogous to the definition of groups. Gyrogroups, both gyrocommutative and non-gyrocommutative, 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 (Möbius) gyrovector spaces form the setting for Beltrami-Klein (Poincaré) ball models of hyperbolic geometry. Finally, novel applications of Möbius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity, are presented.
"synopsis" may belong to another edition of this title.
From the Author:
Geometry is at the foundation of physics. In particular, hyperbolic geometry finds natural home at the foundation of Einstein's special relativity theory,just as Euclidean geometry lies at the foundation of Newtonian physics. By listening to the sounds of relativistic velocities and their compositionby Einstein velocity addition law, analytic hyperbolic geometry emerges, significantly extending Einstein's unfinished symphony.Vector addition is both commutative and associative, giving rise to vector spaces. In full analogy, gyrovector addition is both gyrocommutative andgyroassociative, giving rise to gyrovector spaces. Einstein's addition of relativistic velocities is given by gyrovector addition, just as addition ofNewtonian velocities is given by vector addition. The resulting gyrovector spaces form the algebraic setting for analytic hyperbolic geometry, just asvector spaces form the algebraic setting for analytic Euclidean geometry.The resulting analytic hyperbolic geometry emerges in this book as a new mathematical discipline, fully analogous to the familiar analytic Euclideangeometry. As such, this book uncovers unexpected analogies with classical results, enabling the modern and unfamiliar to be studied in terms of theclassical and familiar.
Review:
?This new book by Ungar is very well-written, with plenty of references and explanatory pictures."
"About this title" may belong to another edition of this title.
- PublisherWorld Scientific Pub Co Inc
- Publication date2005
- ISBN 10 9812564578
- ISBN 13 9789812564573
- BindingHardcover
- Number of pages463
Buy New
Learn more about this copy
US$ 223.49
Shipping:
US$ 46.28
From United Kingdom to U.S.A.
Top Search Results from the AbeBooks Marketplace
Stock Image
Analytic Hyperbolic Geometry: Mathematical Foundations And Applications (Hardcover)
Published by
World Scientific Publishing Co Pte Ltd, Singapore
(2005)
ISBN 10: 9812564578
ISBN 13: 9789812564573
New
Hardcover
Quantity: 1
Seller:
Rating
Book Description Hardcover. Condition: new. Hardcover. 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. Gyrovectors are equivalence classes of directed gyrosegments that add according to the gyroparallelogram law just as vectors are equivalence classes of directed segments that add according to the parallelogram law. 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 prefix stems from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Gyrolanguage turns out to be the language one needs to articulate novel analogies that the classical and the modern in this book share.The scope of analytic hyperbolic geometry that the book presents is cross-disciplinary, involving nonassociative algebra, geometry and physics. As such, it is naturally compatible with the special theory of relativity and, particularly, with the nonassociativity of Einstein velocity addition law. Along with analogies with classical results that the book emphasizes, there are remarkable disanalogies as well. Thus, for instance, unlike Euclidean triangles, the sides of a hyperbolic triangle are uniquely determined by its hyperbolic angles. Elegant formulas for calculating the hyperbolic side-lengths of a hyperbolic triangle in terms of its hyperbolic angles are presented in the book.The book begins with the definition of gyrogroups, which is fully analogous to the definition of groups. Gyrogroups, both gyrocommutative and non-gyrocommutative, 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 (Moebius) gyrovector spaces form the setting for Beltrami-Klein (Poincare) ball models of hyperbolic geometry. Finally, novel applications of Moebius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity, are presented. Presents a gyrovector space approach to analytic hyperbolic geometry, fully analogous to the well-known vector space approach to Euclidean geometry. Including the definition of gyrogroups, this book presents applications of Mobius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity. Shipping may be from our UK warehouse or from our Australian or US warehouses, depending on stock availability. Seller Inventory # 9789812564573
Buy New
US$ 223.49
Convert currency