After a slightly peculiar dream the other night, I find myself suddenly inspired to do numerical simulations in three-dimensional hyperbolic space.
For this to work, I need an equation of line in hyperbolic space --- some $f(p_1, p_2, s)$ where $p_1$ and $p_2$ are points and $s$ is a distance, which projects the geodesic which connects the two points. $s$ needs to be a distance as measured along the geodesic (i.e. measured using the space's internal geometry, rather than measured using the projective model of whetever coordinate system we're using).
Some basic reading up shows that there are multiple different coordinate systems for hyperbolic space, the most common one being the Poincaré ball. However the maths look kind of painful, and knowing which model is the most suitable for my purposes is beyond my experience.
I found a nice paper here which describes a number of coordinate systems and the 'element of line' for each, but unfortunately I don't know if that's what I'm looking for or not... A description of several coordinate systems for hyperbolic spaces
Can anyone point me in the right direction?
Incidentally, as a nice-to-have, I'd quite like to be able to roughly map from Euclidean space to hyperbolic space at small scales, because this allows me to construct geometry inside my space without too much pain. This seems to be possible with the Poincaré projection because it's just polar coordinates in the projected sphere. But I have no idea about the other coordinate systems.
FWIW, I seem to be settling on to the Beltrami-Klein model. It has the property that a geodesic in hyperbolic space maps onto a straight line in the model, which means that calculating intersections is cheap and easy; the only part where the spatial curvature becomes important is calculating the hyperbolic distance along the line where my intersection occurs.
According to Wikipedia the formula for hyperbolic distance for the geodesic between points on the model $p$ and $q$ is this:
$$d(p,q)=\frac{1}{2} \log \frac{ \left| aq \right| \, \left| pb \right| }{ \left| ap \right| \, \left| qb \right| }$$
...where $a$ and $b$ and the points on the model where my line $pq$ intersects the unit circle.
This seems to simplify (complexify?) to:
$$d(p,q) = \sinh^{-1} \frac{\sqrt{|D_{pq}^2-A_{pq}^2|}}{\sqrt{1-D_p^2}\sqrt{1-D_q^2}},$$
where $D_{pq}=|p-q|$ is the Euclidean distance between the points, $D_p=|p|$ and $D_q=|q|$ are the Euclidean distances from the origin, and $A=x_py_q-x_qy_p$ (I haven't figured out what this represents yet).
(Stolen from Convert Euclidean distance to Hyperbolic distance.)
I'm still don't know how curvature maps onto this, as the equation above assume constant curvature of -1 and I may want to change this, but I'll ask a separate question on that.