Super Hyperbolic Law of Cosines: same formula with different content

Abstract

We derive the Laws of Cosines and Sines in the super hyperbolic plane using Minkowski supergeometry and find the identical formulae to the classical case, but remarkably involving different expressions for the trigonometric functions, which include substantial fermionic corrections. In further analogy to the classical case, we apply these results to show that two parallel supergeodesics which are not ultraparallel admit a unique common orthogonal supergeodesic, and we briefly describe aspects of elementary supernumber theory, leading to a prospective analogue of the Gauss product of quadratic forms.