Different models of the hyperbolic line

Point parameter:
Isometry parameter:
This was created for MA243, second year geometry course, Autumn 2023.
This shows various version of the hyperbolic line, and a point P, and another point T(P), where P is determined by a parameter t and T is determined by a parameter γ.
Let's parameterise motions (isometries) of the hyperbolic line H1. These are either translations or reflections. As a topological space, this is two disjoint copies of the real line, one for the translations, one for the reflections. You can't continuously go from a translation to a reflection. We can take several possible modles of H1. H1 is pretty simple to understand, so why bother with several models? Because H2 is more complicated to understand, and we work with the hyperboloid model, and the projection to the plane, but you may be also be familiar with the Poincare disc. And we also draw lines obtained by projection to the x2,x3 plane. And we consider a parametrisation. The H1 case is very similar, and simpler to understand. The little black marks are spaced at unit intervals along the lines, with respect to the various metric.

Point P

Isometry T

The isometry is either a translation or reflection. The matrix describing the isometry as an operator on the hyperbola model has the form [cosh(γ),sinh(γ);sinh(γ),cosh(γ)], for a translation, or [cosh(γ),-sinh(γ);sinh(γ),-cosh(γ)] for a reflection, read row by row.

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