I'm a new guy studying the distance metric in Poincare Disk model. The measure distance between two random points u and v on the Poincare disk is described as follows:
$$ d\left( \mathbf u, \mathbf v \right) = arcosh \left( 1+2\frac{||\mathbf u- \mathbf v||^2}{(1-||\mathbf u||^2)(1-||\mathbf v||^2)}\right)$$
I wnat to modify the distance measure method by adding the bias vector \mathbf r,
$$ d\left( \mathbf u, \mathbf v \right) = arcosh \left( 1+2\frac{||\mathbf u- \mathbf v ||^2}{(1-||\mathbf u+ \mathbf r||^2)(1-||\mathbf v||^2)}\right)$$
The new distance is asymmetry and thus brings an property of direction, which is helpful is my research area.
I'm not confident if this way is reasonalbe. My problem is: Does this measure make sense? Has anyone tried this measure in any areas? I appreciate any analysis and comments. Many thanks.
oh I thoyght of another way to use asymmetry as well as keeping zero distance when u equals v.
$$d\left( \mathbf u, \mathbf v \right) = arcosh \left( 1+2\frac{\|\mathbf u- \mathbf v \|^2 +\|\mathbf u\|^2 -\|\mathbf v\|^2}{(1-\|\mathbf u\|^2)(1-\|\mathbf v\|^2)}\right)$$
Does this make sense?