Abraham Albert Ungar

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Hyperbolic trigonometry is developed and illustrated in this article along lines parallel to Euclidean trigonometry by exposing the hyperbolic trigonometric law of cosines and of sines in the Poincaré ball model of n-dimensional hyperbolic geometry, as well as their application. The Poincaré ball model of 3-dimensional hyperbolic geometry is becoming(More)
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 " gyrolanguage " of this paper one attaches the prefix " gyro " to a classical term to mean the(More)
An involutory decomposition is a decomposition, due to an involution, of a group into a twisted subgroup and a subgroup. We study unexpected links between twisted subgroups and gyrogroups. Twisted subgroups arise in the study of problems in computational complexity. In contrast, gyrogroups are grouplike structures which rst arose in the study of Einstein's(More)
Gyrogroups are generalized groups modelled on the Ein-stein groupoid of all relativistically admissible velocities with their Einstein's velocity addition as a binary operation. Ein-stein's gyrogroup fails to form a group since it is nonassocia-tive. The breakdown of associativity in the Einstein addition does not result in loss of mathematical regularity(More)
The hyperbolic trigonometry, fully analogous to the common Eu-clidean trigonometry, is presented and employed to calculate the hyperbolic triangle defect in the Poincaré ball model of n-dimensional hyperbolic geometry. It is shown that hyperbolic trigonometry allows the hyperbolic triangle defect to be expressed in terms of the triangle hyperbolic side(More)
Some gyrocommutative gyrogroups, also known as Bruck loops or K-loops, admit scalar multiplication, turning themselves into gyrovector spaces. The latter, in turn, form the setting for hyperbolic geometry just as vector spaces form the setting for Euclidean geometry. In classical mechanics the centroid of a triangle in velocity space is the velocity of the(More)