Talk:Coordinate systems for the hyperbolic plane

Use haversine formulas?
In the section on polar coordinates, we currently have this formula for the distance between two points
 * $$\operatorname{dist} (\langle r_1, \theta_1 \rangle, \langle r_2, \theta_2 \rangle) = \operatorname{arcosh} \, \left( \cosh r_1 \cosh r_2 - \sinh r_1 \sinh r_2 \cos (\theta_2 - \theta_1) \right) \,.$$

Or if we use D for the distance,
 * $$\cosh D = \cosh r_1 \cosh r_2 - \sinh r_1 \sinh r_2 \cos (\theta_2 - \theta_1) \,.$$

It would be nice if we could improve this in two ways: (1) make it look more like the Pythagorean theorem from Euclidean geometry which serves the same function, and (2) avoid the loss of significance due to the fact that cosh and cos remain large (of order 1) even when their arguments are small. Well, I recently read about the haversine formula which can help to achieve these objectives. The basic idea is to replace cosh and cos as follows:
 * $$\cosh x = 1 + 2 \sinh^2 \frac{x}{2} $$
 * $$\cos x = 1 - 2 \sin^2 \frac{x}{2} $$

while being clever about the order of replacement and then cancelling out the remaining 1s. For our formula, it works as follows
 * $$\cosh D = \cosh r_1 \cosh r_2 - \sinh r_1 \sinh r_2 \cos (\theta_2 - \theta_1) = \cosh r_1 \cosh r_2 - \sinh r_1 \sinh r_2 + 2 \sinh r_1 \sinh r_2 \sin^2 \frac{\theta_2 - \theta_1}{2} \,$$
 * $$\cosh D = \cosh (r_2 - r_1) + 2 \sinh r_1 \sinh r_2 \sin^2 \frac{\theta_2 - \theta_1}{2} \,$$
 * $$1 + 2 \sinh^2 \frac{D}{2} = 1 + 2 \sinh^2 \frac{r_2 - r_1}{2} + 2 \sinh r_1 \sinh r_2 \sin^2 \frac{\theta_2 - \theta_1}{2} \,$$
 * $$\sinh^2 \frac{D}{2} = \sinh^2 \frac{r_2 - r_1}{2} + \sinh r_1 \sinh r_2 \sin^2 \frac{\theta_2 - \theta_1}{2} \,$$
 * $$D = 2 \operatorname{arsinh} \sqrt{ \sinh^2 \frac{r_2 - r_1}{2} + \sinh r_1 \sinh r_2 \sin^2 \frac{\theta_2 - \theta_1}{2} } \,.$$

Applying the same idea to our formula for distance in Lobachevsky coordinates gives
 * $$\cosh D = \cosh y_1 \cosh (x_2 - x_1) \cosh y_2 - \sinh y_1 \sinh y_2 \,$$
 * $$\cosh D = \cosh y_1 \cosh y_2 - \sinh y_1 \sinh y_2 + 2 \cosh y_1 \cosh y_2 \sinh^2 \frac{x_2 - x_1}{2} \,$$
 * $$\cosh D = \cosh (y_2 - y_1) + 2 \cosh y_1 \cosh y_2 \sinh^2 \frac{x_2 - x_1}{2} \,$$
 * $$1 + 2 \sinh^2 \frac{D}{2} = 1 + 2 \sinh^2 \frac{y_2 - y_1}{2} + 2 \cosh y_1 \cosh y_2 \sinh^2 \frac{x_2 - x_1}{2} \,$$
 * $$\sinh^2 \frac{D}{2} = \sinh^2 \frac{y_2 - y_1}{2} + \cosh y_1 \cosh y_2 \sinh^2 \frac{x_2 - x_1}{2} \,$$
 * $$\sinh^2 \frac{D}{2} = \cosh y_1 \cosh y_2 \sinh^2 \frac{x_2 - x_1}{2} + \sinh^2 \frac{y_2 - y_1}{2} \,$$
 * $$D = 2 \operatorname{arsinh} \sqrt{ \cosh y_1 \cosh y_2 \sinh^2 \frac{x_2 - x_1}{2} + \sinh^2 \frac{y_2 - y_1}{2} } \,$$

Unfortunately, these formulas are a little messier to write down than the original formulas. So do you think it is worth changing them? JRSpriggs (talk) 03:50, 18 June 2016 (UTC)

Missing information on axial coordinates
Since we have provided a distance formula, metric tensor, and an equation for straight lines for polar coordinates, Lobachevsky coordinates and horocycle-based coordinates, the reader may wonder why we have not provided this information for axial coordinates. It appears to me that this information would be much more complicated for axial coordinates. This maybe for two reasons: axial coordinates only have a small set of symmetries (an eight element group) while the others have large groups of symmetries (continuous and thus infinite), the curves of constant x and constant y are oblique in axial coordinates while they are perpendicular to each other in the other coordinate systems. In particular, the metric in axial coordinates has cross-terms while the metrics in the other coordinate systems do not have cross-terms.

To illustrate this problem, I tackled the easiest of the three formulas, the metric tensor. After a lot of work converting the metric from polar coordinates to axial coordinates, I got what I hope is the correct formula:
 * $$ (ds)^2 = \frac{(1 - \tanh^2 x) (1 - \tanh^2 y)}{(1 - \tanh^2 x - \tanh^2 y)^2} ( (dx)^2 + (dy)^2 - (\tanh x \, dx - \tanh y \, dy)^2 ) \,.$$

You see the difficulty? JRSpriggs (talk) 13:06, 1 July 2016 (UTC)