Fig. 3

The “composite distance” to a point of the boundary. Definition of the quantity \(\langle x, b \rangle\) if x is a point of \(\mathbb{D}\) and b a point of its boundary: \(\xi(b,x)\) is the horocycle through x which is tangent to the boundary at b, and \(\Delta(b, x)\) is the segment joining the origin O to the point on \(\xi(b,x)\) which is diametrically opposite b; the number \(\langle x, b \rangle\) is, up to a sign, the hyperbolic length of this segment