Harmonic analysis on the proper velocity gyrogroup

Ptolemy’s theorem in Euclidean geometry, named after the Greek astronomer and mathematician Claudius Ptolemy, is well known. We translate Ptolemy’s theorem from analytic Euclidean geometry into the Poincaré ball model of analytic hyperbolic geometry, which is based on the Möbius addition and its associated symmetry gyrogroup. The translation of Ptolemy’s theorem from Euclidean geometry into hyperbolic geometry is achieved by means of hyperbolic trigonometry, called gyrotrigonometry, to which the Poincaré ball model gives rise, and by means of the duality of trigonometry and gyrotrigonometry.

Section: Möbius Addition and Scalar Multiplicationmentioning

confidence: 99%

“…Applying the law of gyrocosines ( 13) to the O-gyrovertex gyroangle α = ∠AOB of gyrotriangle ABO, shown in Fig. 3, yields (15) cos…”

Section: Ptolemy's Theorem In the Poincaré Ball Model Of Hyperbolic G...mentioning

confidence: 99%

The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry

Ungar

2023

“…There appeared a theory that may be called gyrogeometry, based on nice algebraic properties of the aforementioned operations and the concepts of gyrogroups, gyrovector spaces, gyrotrigonometry, and gyrogeometric objects [21][22][23][24][25][26][27][28][29][30]. This theory has been extended to the space of matrices with indefinite scalar products, which is closely related to the phenomenon of entanglement in quantum physics [32][33][34], and to harmonic analysis [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

“…Find necessary and sufficient conditions for binary operations to be isomorphic to Euclidean addition. 7. Find necessary and sufficient conditions for binary operations to be isomorphic to Einstein addition.…”

Section: Introductionmentioning

confidence: 99%

Isomorphism of Binary Operations in Differential Geometry

Barabanov¹

2020

We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from associated metric tensors that have a canonical form. Necessary and sufficient conditions for canonical metric tensors to generate binary operations are found. A definition of algebraic isomorphism of binary operations is given. Necessary and sufficient conditions for binary operations to be isomorphic are provided. It is proved that every algebraic automorphism gives rise to isomorphism of corresponding gyrogroups. Necessary and sufficient conditions in terms of metric tensors for binary operations to be isomorphic to Euclidean addition are given. The problem of binary operations to be isomorphic to Einstein addition is also solved in terms of necessary and sufficient conditions. We also obtain necessary and sufficient conditions for binary operations having the same function-parameter in the canonical representation of metric tensors to be isomorphic.

“…Gyrogroups share remarkable analogies with groups studied, for instance, in References [27][28][29][30][31][32]. Applications of gyrogroups in harmonic analysis are found in References [17,23,33]. For other interesting studies of gyrogroups see References [34][35][36][37][38][39].…”

mentioning

confidence: 99%

Binary Operations in the Unit Ball: A Differential Geometry Approach

Barabanov¹,

Ungar²

2020

Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n ∈ N , and discover the properties that qualify these operations to the title addition despite the fact that, in general, these binary operations are neither commutative nor associative. The binary operation of the Beltrami-Klein ball model of hyperbolic geometry, known as Einstein addition, and the binary operation of the Beltrami-Poincaré ball model of hyperbolic geometry, known as Möbius addition, determine corresponding metric tensors in the unit ball. For a variety of metric tensors, including these two, we show how binary operations can be recovered from metric tensors. We define corresponding scalar multiplications, which give rise to gyrovector spaces, and to norms in these spaces. We introduce a large set of binary operations that are algebraically equivalent to Einstein addition and satisfy a number of nice properties of this addition. For such operations we define sets of gyrolines and co-gyrolines. The sets of co-gyrolines are sets of geodesics of Riemannian manifolds with zero Gaussian curvatures. We also obtain a special binary operation in the ball, which is isomorphic to the Euclidean addition in the Euclidean n-space.