Regiomontanus Angle Optimization for Circular Paths

The classical Regiomontanus problem asks for the position that maximizes the angle subtended by a fixed line segment, a problem originating in Euclidean geometry and observational astronomy. This work extends the problem to observers constrained to move along a circular path. The solution uses the geometric fact that the optimal viewpoint occurs precisely where the trajectory is tangent to an isoangle circle. This framework unifies configurations both exterior and interior to the circular path. The analytical solution is validated against direct numerical optimization, with agreement exceeding ten decimal places, and the subtended angle is shown to attain a unique maximum along each half of the circular trajectory. In the limit as the radius becomes large, the analytical result reduces to the classical linear case. Beyond its historical interest, a potential extension illustrates how this simple geometric condition can govern optimal viewing and highlights connections to modern imaging and vision systems.