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- Abraham A Ungar
- 2001

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)

- Abraham A Ungar
- 2006

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)

- Tuval Foguel, Abraham A Ungar, Typeset By, A M S-T
- 2007

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)

- Tuval Foguel, Abraham A Ungar
- 2000

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)

- Abraham A Ungar
- 2004

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)

- Abraham A Ungar
- 1998

A ABSTRACT new form of the Hyperbolic Pythagorean Theorem, which has a striking t e intuitive appeal and offers a strong contrast to its standard form, is presented. I xpresses the square of the hyperbolic length of the hypotenuse of a hyperbolic s o right angled triangle as the "Einstein sum" of the squares of the hyperbolic length f the other two sides,… (More)

- Abraham A Ungar
- 2010

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)