barycentre in hyperbolic geometry :

Question

In euclidean geometry we know the formulas for midpoint and barycentre of a finite set of points, so can we find similar formulas in hyperbolic geometry ? In the Klein disk, Ungar cited in his book "A gyrovector space approach to Hyperbolic geoemtry" that the barycentroid of $(u,v,w)$ is given by $$m_{uvw}= \frac{\gamma_uu+\gamma_v v+\gamma_w w}{\gamma_u+\gamma_v +\gamma_w } $$ with $\gamma_u=\frac{1}{\sqrt{1-\|u\|2}}$ the Lorentz factor. I am looking for simple formulas without using $\gamma_u$, so is this possible?

Answer 1

Converting from the Klein disk to the hyperboloid model maps a point $p = (x, y)$ to $p_H = (\gamma_px, \gamma_py, \gamma_p)$. So in the hyperboloid model of hyperbolic geometry, this formula simplifies to

$$m_{uvw} = \frac{u+v+w}{\|u+v+w\|}$$

where $\|(x,y,t)\| := \sqrt{|x^2+y^2-t^2|}$ denotes the Minkowski "norm".