# Valid metric on a hyperbolic space

Note: originally posted on math.SE.

I'm looking at the distance that's defined in this paper on Poincaré Embeddings:

$$d(\mathbf{u}, \mathbf{v}) = \operatorname{arccosh} \left(1 + 2\frac{\left\| \mathbf{u} - \mathbf{v} \right\|^2}{(1 - \left\| \mathbf{u} \right\|^2)(1 - \left\| \mathbf{v} \right\|^2)} \right)$$

for $$\mathbf{u}, \mathbf{v} \in \mathcal{B}^d$$ using the Poincaré ball model of hyperbolic space.

Does this define a valid metric? It's easy to see that

• $$d(\mathbf{u}, \mathbf{v}) \geq 0$$ since $$\left\| \mathbf{u} \right\| \leq 1$$ and $$\left\| \mathbf{v} \right\| \leq 1$$ and so the rhs inside the parentheses is positive ($$\operatorname{arccosh}(x)$$ is an increasing function for $$x \geq 0$$),
• $$\mathbf{u} = \mathbf{v}$$, then $$d(\mathbf{u}, \mathbf{v}) = 0$$, and
• $$d(\mathbf{u}, \mathbf{v}) = d(\mathbf{v}, \mathbf{u})$$

However I'm having a hard time trying to prove the triangle inequality in this case. I've tried using the logarithmic form but that didn't get me anywhere. I also found the identity $$\operatorname {arcosh} u\pm \operatorname {arcosh} v=\operatorname {arcosh} \left(uv\pm {\sqrt {(u^{2}-1)(v^{2}-1)}}\right)}$$ under "Addition Formulae (sic.)" on Wikipedia but that also seemed to be a dead end. Am I missing something obvious?

• While this paper uses the Poincaré ball model and is even titled after it, they really should be using the Minkowski hyperboloid model instead (and Poincaré only for presentation). I suppose they chose the Poincaré ball model because they did not know much about computational hyperbolic geometry. See the followup paper by the same authors, it changes to the Minkowski hyperboloid model, and it works much better. Also it is much easier to understand and to work with. Mar 4, 2019 at 11:53
• Interesting ... thanks. Which paper are you referring to?
– tdc
Mar 4, 2019 at 11:58
• This one: arxiv.org/pdf/1806.03417.pdf (they call it the Lorentz model, but it is more commonly called the hyperboloid model). Mar 4, 2019 at 12:02
• Agreed! Much simpler ...
– tdc
Mar 4, 2019 at 23:28
• Also, it's good to have some context around hyperbolic space. The formula for distance follows from more geometrically intuitive descriptions of geodesics and Riemannian metric.
– Neal
Mar 6, 2019 at 3:12