A Degenerate Spherical Triangle

Abstract

This paper investigates the boundary limits of non-Euclidean polygons, focusing specifically on the structural transition of a three-sided manifold into a degenerate state. On a spherical surface, the summation of interior angles is fundamentally tied to bounded surface area, always exceeding the classical Euclidean limit of 180°. Here, we analyze the specific limiting case where the interior angular sum reaches a maximum structural boundary of 360°. Rather than retaining its properties as a valid spherical triangle or transitioning into a two-sided spherical lune, the shape undergoes a complete topological reduction. We show that at this precise threshold, internal area drops to zero, and the independent coordinate positions of the three original paths collapse into a single, continuous semicircular trajectory. This boundary condition is formally defined as a degenerate spherical triangle, providing a clear physical demonstration of spatial collapse within curved manifolds.