Proof of the Properties of D

Properties of D
For z,w ε D,
  • D(0,w) = |w|,
  • D(z,w) ≥ 0,
  • D(z,w) = 0 if and only if z = w,
  • D(z,w) = D(w,z).

Proof
We have D(z,w) = |z-w|/|w*z-1|.
The first three are elementary consequences.
For the fourth, observe that
|w-z| = |-(z-w)| = |z-w|, and
|z*w-1| = |(z*w-1)*| = |w*z-1|, so that
D(w,z) = |w-z|/|z*w-1| = |z-w|/|w*z-1| = D(z,w).
return to hyperbolic circles