Special Relativity (SR) assumes that velocity vectors, displacement vectors and direction vectors coincide, since they are projected onto a flat analytical space (Minkowski space) to interpret inertial relative motion. In contrast, Arnold Sommerfeld’s 1909 spherical model interprets velocity composition on a spherical surface, where displacement and constant direction vectors are fundamentally distinct. This paper demonstrates that because Einstein and Sommerfeld use different surfaces to project their interpretation of the Michelson-Morley interferometry, their predictions for relativistic effects diverge. Specifically, while Einstein’s model necessitates Lorentz contraction and time dilation, Sommerfeld’s spherical formulation predicts no such distortions. Notably, compatibility between SR and Sommerfeld’s spherical model exists only in the special case of two right-angled triangles. By extension, any hyperbolic model that claims compatibility with SR must also be restricted to this special case. This work highlights a fundamental geometric conflict between Special Relativity and Sommerfeld’s alternative formulation, warranting further examination of the geometric nature of inertial motion.