Inversion

Move the point M in the interior of the disc and to the boundary of the disc to see that its inverse image point M' behaves in the same manner.

The one side of the space is mapped to the other and vice versa just as happens in reflection about a line in the Euclidean plane. Since the Mobius transformation is conformal and the mirror d-line cuts the horizon at 90 degrees we see that the boundary one one side is mapped exactly onto the boundary of the other side of the d-line.