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The hyperbolic triangle centroid

Abraham A. Ungar — 2004

Commentationes Mathematicae Universitatis Carolinae

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 center of momentum of three massive objects with equal masses located at the triangle vertices. Employing gyrovector space techniques we find...

Möbius gyrovector spaces in quantum information and computation

Abraham A. Ungar — 2008

Commentationes Mathematicae Universitatis Carolinae

Hyperbolic vectors, called gyrovectors, share analogies with vectors in Euclidean geometry. It is emphasized that the Bloch vector of Quantum Information and Computation (QIC) is, in fact, a gyrovector related to Möbius addition rather than a vector. The decomplexification of Möbius addition in the complex open unit disc of a complex plane into an equivalent real Möbius addition in the open unit ball 𝔹 2 of a Euclidean 2-space 2 is presented. This decomplexification proves useful, enabling the resulting...

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