The Geometry of Collapse: Structural Indeterminacy, Kalimuthu's Extremes, and Boundary Transitions in Degenerate Antipodal Spherical Triangles

Abstract

This paper analyzes the geometric breakdown of spherical triangles at antipodal boundary conditions. For the degenerate configuration a=90°, b=90°, c=180°, we show that the spherical Law of Cosines collapses to the indeterminate identity 0=0 and the Law of Sines to the undefined form 0/0. We provide a formal justification by demonstrating that the system possesses a continuous free parameter - the longitude λ of the equatorial vertex - yielding a continuum of valid configurations with identical side lengths. This configuration is presented as a corollary of Kalimuthu's theorem, which permits a closed spherical triangle with angle sum 360° via a straight-angle vertex. Despite angular indeterminacy, Girard's theorem confirms the area remains stable and exactly 1/4 of the sphere (πR²), bridging classical triangles and multi-angle lunes.