Distance Function:

\[
   f_r(X)=\frac{1}{2} \ln{ \left(\frac{XP}{XQ}\right)  }
\]

Distance:

\[
   d_K(X,Y)=
   \frac{1}{2} \ln{ \left(\frac{XP}{XQ} \cdot \frac{YQ}{YP}\right)  }
\]

<CENTER>
   @Image Klein.Dist.svg height=100px
</CENTER>

\[
   XP=t||\underline{e}_Q-\underline{e}_P||,
   \quad
   XQ=(1-t)||\underline{e}_Q-\underline{e}_P||
\]

\[
   f_r(t)=\frac{1}{2}
      \ln{\left(
         \frac{t}{1-t}
      \right)}
\]

Compare, Inverse Tangent Hyperbolic:

\[
   Artanh~x=\frac{1+x}{1-x},
     ~-1 \lt x \lt 1
\]