## Complex distance between two hyperbolic lines

An oriented hyperbolic line in hyperbolic $3$-space $H^3$ is uniquely determined by its “initial” ideal point (by ideal point, I mean a point on the sphere at infinity $S^2_\infty$) and its “final” ideal point. We can thus write, by abuse of language, that an oriented hyperbolic line $L$ “is”: $L = (a, b)$, where $a$, resp. $b$, is its initial, resp. final, point. Now suppose we are given two oriented hyperbolic lines, say $L_i = (a_i, b_i)$, for $i =…