Spherical curves and quadratic relationships for special functions

We show that the position vector of any 3-space curve lying on a sphere satisfies a third-order linear (vector) differential equation whose coefficients involve a single arbitrary function A(s) . By making various identifications of A(s) , we are led to nonlinear identities for a number of higher transcendental functions: Bessel functions, Horn functions, generalized hypergeometric functions, etc . These can be considered natural geometrical generalizations of sin 2 t + cos 2 t = 1. We conclude with some applications to the theory of splines.