Angle of parallelism

Spherical metrics in general relativity

Published in Maricel Agop, Ioan Merches, Operational Procedures Describing Physical Systems, 2018

This metric reduces to that of Poincare in case Ω1 =0 which, as Barbilian noticed, defines the variable φ as the ‘angle of parallelism’ of the hyperbolic plane (the connection). In fact, recalling that in modern terms (du/v) represents the connection form of the hyperbolic plane, the equations (2.188) then represent a general Bäcklund transformation in that plane [14].

Dynamic hyperbolic geometry: building intuition and understanding mediated by a Euclidean model

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Journal Information

Published in International Journal of Mathematical Education in Science and Technology, 2018

Luis Moreno-Armella, Corey Brady, Rubén Elizondo-Ramirez

An implication of this equation is that the angle of parallelism is always strictly less than 90°. This is quite dissimilar from what we have in Euclidean geometry, where the angle of parallelism is always 90° and is independent of the length d. (This dependence of angle on distance is also the root cause for why we cannot have similar, non-congruent triangles in the hyperbolic case.) A space in which these things could hold would be hostile to intuition, to say the least. Imagine the intellectual effort that those pioneers had to display in order to find a model for a geometry whose propositions insinuated a strange world, never seen, never imagined before!

Angle of parallelism

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Spherical metrics in general relativity

Dynamic hyperbolic geometry: building intuition and understanding mediated by a Euclidean model