I'm following the work from the paper, Poincaré Embeddings for Learning Hierarchical Representations which uses the Poincaré ball model of hyperbolic space. In equation [1] they give the distance between any two points as
$$ d(\textbf{u},\textbf{v}) = \text{arcosh} \left ( 1 + 2\frac{||\textbf{u}-\textbf{v}||^2}{(1-||\textbf{u}||^2)(1-||\textbf{v}||^2)} \right ) $$
here $||\cdot||$ is the Euclidean norm and the vectors are constrained to an open unit ball in $d$ dimensions, $\textbf{u}, \textbf{v} \in \{ \textbf{x} \in \mathcal{R}^d ; ||\textbf{x}||^2 < 1 \}$. So far, so good.
I'd like to consider the notion of an "angle" between two points $\textbf{a}, \textbf{b}$ in this space. By fixing the center $\textbf{c}$ as a special point, one could use the standard angle in Euclidean space defined by the line segments $\textbf{a} \rightarrow \textbf{c} \rightarrow \textbf{b}$, but this doesn't seem natural since the entire nature of the metric is different. Is there some notion of an angle that makes more sense?
I think you want the hyperbolic Law of Cosines. It reads this way: $$ \cosh(\overline{\mathbf a\mathbf b})= \cosh(\overline{\mathbf a\mathbf c}) \cosh(\overline{\mathbf c\mathbf b})- \sinh(\overline{\mathbf a\mathbf c}) \sinh(\overline{\mathbf c\mathbf b})\cos(\angle\mathbf a\mathbf c\mathbf b)\,, $$ and of course you want to solve for the cosine of the angle at the vertex of the triangle.