Question about hyperbolic space and solving for vectors

I’m writing a C++ library for hyperbolic space, inspired by HyperRogue.

I’m representing vectors using the Minkowski hyperboloid, [;x_0^2 – x_1^2 – x_2^2 = 1;]

Now, I’m working on the puzzle of drawing lines between two points represented this way. Wikipedia has given a vague hint:

Given the basis vectors [;u, v;] of a plane, then [;ucosh(w) + vsinh(w);], where [;w in mathbb{R};] should be on the geodesic, so long as the following terms are met:

  1. [;B(u, u) = 1;]
  2. [;B(v, v) = -1;]
  3. [;B(u, v) = 0;]

Where [;B(a, b) = a_0b_0 – a_1b_1 – a_2b_2;]

I would like to find the line between two points, lets say [;o, p;]. Because these two points are on the hyperboloid, they by definition meet the 1st criteria. So, let’s set [;u = o;]

Now the task is finding [;v;], where [;v in Span[o, p];].

I have been attempting this for the past two or three hours and have made no progress since my math education isn’t great.

Please help me with this cause it’s pissing me off :/

submitted by /u/Plazmotech
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Nevin Manimala

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