## Analytic Hyperbolic Geometry: Mathematical Foundations and ApplicationsThis 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 OC gyrolanguageOCO of the book one attaches the prefix OC gyroOCO 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 (MAbius) gyrovector spaces form the setting for BeltramiOCoKlein (Poincar(r)) ball models of hyperbolic geometry. Finally, novel applications of MAbius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity, are presented." |

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### Contents

1 | |

21 | |

Gyrocommutative Gyrogroups | 49 |

Gyrogroup Extension | 87 |

Gyrovectors and Cogyrovectors | 117 |

Gyrovector Spaces | 137 |

### Common terms and phrases

addition law algebra analogies analytic hyperbolic geometry automorphism barycentric coordinates cancellation law coaddition cogyroangle cogyroline commutative Definition density matrix Einstein addition Einstein gyrovector space Einstein velocity addition Einsteinian equal equivalent Euclidean geometry following theorem fully analogous given gives rise groupoid gyr[a gyr[u gyration gyro gyroangle gyroautomorphic inverse gyrobarycentric coordinates gyroboost gyrocentroid gyrocommutative gyrogroup gyrocommutative law gyrogroup cooperation gyrogroup G gyroline gyrometric gyromidpoint gyroparallelepiped gyropolygonal gyrorays gyrosegment gyrotriangle gyrotriangle ABC gyrovec gyrovector translation Hence hyperbolic geometry identity inner product space left cancellation left gyrotranslations Lemma Let G loop property Lorentz boost Lorentz transformation Mobius addition Mobius gyrovector plane Mobius gyrovector space model of hyperbolic parallel transport parallelepiped parameter points Proof proper velocity qubit relativistic mass Riemannian line element right cancellation rooted gyrovector scalar multiplication shown in Fig side gyrolengths space G spacetime special relativity Thomas precession triangle Ungar vector space velocity addition vertices

### Popular passages

Page 1 - Euclid parallel postulate, given a line L and a point P not on L there is one and only one line L' which contains P and is parallel to L.

Page 2 - Lobachevsky did not attain any wide recognition for many years, and it was only after Baltzer had called attention to them in 1867 that non-Euclidean geometry began to be seriously accepted and studied.

Page 1 - ... 3. Attempts to prove the parallel-postulate. It seems impossible to suppose that Euclid ever imagined this to be self-evident, yet the history of the theory of parallels is full of reproaches against the lack of self-evidence of this