## Different models of the hyperbolic line

(sinh(t),cosh(t)
sinh(t)
sinh(t)/(1+cosh(t))
Point parameter:
Isometry parameter:
translation/reflection
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

• t: You can control the point t, either with the slider or the yellow dot on the cyan line. This point can be considered as on the real line with the usual Euclidean metric. This point determines a point...
• (sinh(t),cosh(t)) (if the horizontal coordinate is given first; we actually tend to give the vertical coordinate first). This map is an isometry from the real line with the Euclidean metric to the hyperbola with the hyperbolic metric. This is the pink dot on the red curve, which represents the hyperbola model of the hyperbolic line. This point projects vertically to a point
• sinh(t), marked by a green dot on the green line. This line is another copy of the reals, but this map is not an isometry. The point (sinh(t),cosh(t)) also maps to a point...
• sinh(t)/(1+cosh(t)), which is on the orange interval, which is a one dimensional version of the Poincare disc. This map is via projection from the point (0,-1).

#### 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.

Not written for touch screens; the yellow dot can be moved via mouse on a computer; otherwise use the sliders.