To derive this equation, we use what we know about perpendicular
circles. Any Poincaré line, as part of a Möbius circle, has an
equation of the form for some complex
and real
with
. The edge circle has equation
, so the first circle is perpendicular to the edge whenever
, i.e., whenever
. The equation becomes
and the condition on the coefficients becomes
. We may assume that
is non-negative (mulitply the equation through by
if necessary), so if we take square roots, the condition becomes
.