This is a response to Will Sawin's comment "What algebraic structure would you place on $\mathbb{H}^2$?"
On $\mathbb{R}^n$ there is a vector space structure where
1) The metric is given by $d(\mathbf{u},\mathbf{v})=\|\mathbf{u}-\mathbf{v}\|$
and
2) given three points $U,V,W$ the angle $U\hat WV$ satisfies $\cos \theta = \frac{(-W+U)\cdot(-W+V)}{\|-W+U\|\|-W+V\|}$
Analogously, for some models of $\mathbb{H}^n$ with points identified with a subset of $\mathbb{R}^n$ there is a vector-like structure but with a noncommutative, nonassociative binary operation $\oplus$ where
1) The metric is given by $d(\mathbf{u},\mathbf{v})=\|\mathbf{u} \ominus \mathbf{v} \| $
and
2) given three points $U,V,W$ the angle $U\hat WV$ satisfies $\cos \theta = \frac{(-W\oplus U)\cdot(-W\oplus V)}{\|-W\oplus U\|\|-W\oplus frac{(\ominus W\oplus U)\cdot(\ominus W\oplus V)}{\|\ominus W\oplus U\|\|\ominus W\oplus V\|}$
$\| \|$ and $\cdot $ are the vector norm and dot product inherited from $\mathbb{R}^n$.
$\ominus a$ denotes the left inverse of a.
$a\ominus b$ denotes $a\oplus (\ominus b)$.
Note the use of the trig function "cos" even though this hyperbolic geometry.
For the Beltrami-Klein model the binary operation $\oplus$ is given by the formula for relativistic velocity-addition:.
