The Orloj

The Astronomical Clock of Prague

Show[image = Import["LargeDial.jpg"]];

The 24-hour dial of the Orloj or Astronomical Clock was created by projecting the celestial sphere onto the plane stereographically as follows. We use the north celestial pole (Overscript[P, ⇀]) = {0, 0, 1} as the point of projection, and define the target plane as z = -1. The equation projecting a point (Overscript[r, ⇀]) on the sphere onto the point (Overscript[q, ⇀]) on the plane can be found by solving the two-point equation of a line for (Subscript[q, x]) and (Subscript[q, y]), substituting in the coordinates of (Overscript[P, ⇀]), and setting (Subscript[q, z]) to -1.

Solve[(((Subscript[P, x]) - (Subscript[q, x]))/((Subscript[P, x]) - (Subscript[r, x]))) == (((Subscript[P, y]) - (Subscript[q, y]))/((Subscript[P, y]) - (Subscript[r, y]))) == (((Subscript[P, z]) - (Subscript[q, z]))/((Subscript[P, z]) - (Subscript[r, z]))), {(Subscript[q, x]),
          (Subscript[q, y])}] /. {(Subscript[P, x]) -> 0, (Subscript[P, y]) -> 0,
        (Subscript[P, z]) -> 1, (Subscript[q, z]) -> -1}

Applying the stereographic projection to points on the celestial sphere gives the main curves on the Orloj's dial, as depicted here.