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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)

We show that the algebra of the group ËÄ´¾ µ naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the Lorentz group and its underlying hyperbolic geometry. The superiority of the use of the gyrogroup formalism over the use of the ËÄ´¾ µ formalism for dealing with the Lorentz group in some cases is indicated by (i) the validity… (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)

- Tuval Foguel, Abraham A. Ungar
- 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)

- Abraham Albert Ungar
- The American Mathematical Monthly
- 2008

- 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)

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 Albert Ungar
- A Gyrovector Space Approach to Hyperbolic…
- 2009

- Abraham Albert Ungar
- Quantum Information & Computation
- 2002