Analytic Hyperbolic Geometry Mathematical Foundations And Applications
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Analytic Hyperbolic Geometry
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Author : Abraham A. Ungar
language : en
Publisher: World Scientific
Release Date : 2005
Analytic Hyperbolic Geometry written by Abraham A. Ungar and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Mathematics categories.
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 (Mbius) gyrovector spaces form the setting for Beltrami-Klein (Poincar) ball models of hyperbolic geometry. Finally, novel applications of Mbius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity, are presented.
Analytic Hyperbolic Geometry And Albert Einstein S Special Theory Of Relativity Second Edition
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Author : Abraham Albert Ungar
language : en
Publisher: World Scientific
Release Date : 2022-02-22
Analytic Hyperbolic Geometry And Albert Einstein S Special Theory Of Relativity Second Edition written by Abraham Albert Ungar and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-02-22 with Mathematics categories.
This book presents a powerful way to study Einstein's special theory of relativity and its underlying hyperbolic geometry in which analogies with classical results form the right tool. The premise of analogy as a study strategy is to make the unfamiliar familiar. Accordingly, this book introduces the notion of vectors into analytic hyperbolic geometry, where they are called gyrovectors. Gyrovectors turn out to be equivalence classes that add according to the gyroparallelogram law just as vectors are equivalence classes that add according to the parallelogram law. In the gyrolanguage of this book, accordingly, one prefixes a gyro to a classical term to mean the analogous term in hyperbolic geometry. As an example, the relativistic gyrotrigonometry of Einstein's special relativity is developed and employed to the study of the stellar aberration phenomenon in astronomy.Furthermore, the book presents, for the first time, the relativistic center of mass of an isolated system of noninteracting particles that coincided at some initial time t = 0. It turns out that the invariant mass of the relativistic center of mass of an expanding system (like galaxies) exceeds the sum of the masses of its constituent particles. This excess of mass suggests a viable mechanism for the formation of dark matter in the universe, which has not been detected but is needed to gravitationally 'glue' each galaxy in the universe. The discovery of the relativistic center of mass in this book thus demonstrates once again the usefulness of the study of Einstein's special theory of relativity in terms of its underlying hyperbolic geometry.
Analytic Hyperbolic Geometry Mathematical Foundations And Applications
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Author : Abraham Albert Ungar
language : en
Publisher: World Scientific
Release Date : 2005-09-05
Analytic Hyperbolic Geometry Mathematical Foundations And Applications written by Abraham Albert Ungar and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-09-05 with Mathematics categories.
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.
Sources Of Hyperbolic Geometry
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Author : John Stillwell
language : en
Publisher: American Mathematical Soc.
Release Date : 1996
Sources Of Hyperbolic Geometry written by John Stillwell and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with Mathematics categories.
Presents the papers of Beltrami, Klein, and Poincare that brought hyperbolic geometry into the mainstream of mathematics.
A Gyrovector Space Approach To Hyperbolic Geometry
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Author : Abraham Ungar
language : en
Publisher: Morgan & Claypool Publishers
Release Date : 2009-03-08
A Gyrovector Space Approach To Hyperbolic Geometry written by Abraham Ungar and has been published by Morgan & Claypool Publishers this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-03-08 with Technology & Engineering categories.
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. Table of Contents: Gyrogroups / Gyrocommutative Gyrogroups / Gyrovector Spaces / Gyrotrigonometry
Hyperbolic Geometry
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Author : Birger Iversen
language : en
Publisher: CUP Archive
Release Date : 1992-12-17
Hyperbolic Geometry written by Birger Iversen and has been published by CUP Archive this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992-12-17 with Mathematics categories.
Although it arose from purely theoretical considerations of the underlying axioms of geometry, the work of Einstein and Dirac has demonstrated that hyperbolic geometry is a fundamental aspect of modern physics. In this book, the rich geometry of the hyperbolic plane is studied in detail, leading to the focal point of the book, Poincare's polygon theorem and the relationship between hyperbolic geometries and discrete groups of isometries. Hyperbolic 3-space is also discussed, and the directions that current research in this field is taking are sketched. This will be an excellent introduction to hyperbolic geometry for students new to the subject, and for experts in other fields.
Old And New Topics In Geometry
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Author : Franz Rothe
language : en
Publisher: Litprime Solutions
Release Date : 2022-09-22
Old And New Topics In Geometry written by Franz Rothe and has been published by Litprime Solutions this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-09-22 with Education categories.
A decade long experience of teaching the course "Fundamental of Geometry", many notes for exercises, and endless extra reading are the bases for this bulky work. The online manuscript already includes many topics with many exercises including solutions and hundreds a elaborate computer generated drawings. The first volume begins with Hilbert's axioms from the Foundations of Geometry, and goes on to projective, neutral and basic Euclidean geometry. The present second volume deals with many more advanced topics form Euclidean geometry and contains a long treatise about hyperbolic geometry. Here the disk models of Poincar\'e and Klein are used to do a lot of constructions, using straightedge and compass from the background Euclidean geometry. Too, Hilbert's axiomatic approach based on the asymptotic rays, is explained from the beginning up to the reconstruction of the Poincar\'e disk model. The last section gives a short course on Gauss' differential geometry and the pseudo sphere.
Foundations Of Hyperbolic Manifolds
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Author : John G. Ratcliffe
language : en
Publisher: Springer
Release Date : 2019-11-07
Foundations Of Hyperbolic Manifolds written by John G. Ratcliffe and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-11-07 with Mathematics categories.
This heavily class-tested book is an exposition of the theoretical foundations of hyperbolic manifolds. It is a both a textbook and a reference. A basic knowledge of algebra and topology at the first year graduate level of an American university is assumed. The first part is concerned with hyperbolic geometry and discrete groups. The second part is devoted to the theory of hyperbolic manifolds. The third part integrates the first two parts in a development of the theory of hyperbolic orbifolds. Each chapter contains exercises and a section of historical remarks. A solutions manual is available separately.
Analytical And Geometric Aspects Of Hyperbolic Space
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Author : D. B. A. Epstein
language : en
Publisher: CUP Archive
Release Date : 1987-03-19
Analytical And Geometric Aspects Of Hyperbolic Space written by D. B. A. Epstein and has been published by CUP Archive this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987-03-19 with Mathematics categories.
This work and its companion volume form the collected papers from two symposia held at Durham and Warwick in 1984. Volume I contains an expository account by David Epstein and his students of certain parts of Thurston's famous mimeographed notes. This is preceded by a clear and comprehensive account by S. J. Patterson of his fundamental work on measures on limit sets of Kleinian groups.
Analytic Hyperbolic Geometry And Albert Einstein S Special Theory Of Relativity Second Edition
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Author : Abraham Albert Ungar
language : en
Publisher:
Release Date : 2022
Analytic Hyperbolic Geometry And Albert Einstein S Special Theory Of Relativity Second Edition written by Abraham Albert Ungar and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022 with Electronic books categories.