The Euler parameters of the rotation of a rigid body in space, termed a spherical displacement, may be used to define a kinematic mapping of the spherical displacement to a point on a hypersphere of unit radius. Thus the differential geometry of curves on a hypersphere reflects instantaneous properties of spherical motion. This paper presents the Frenet reference frame, the Frenet equations, and the geodesic curvature and torsion functions which are used to analyze and characterize curves on a hypersphere. Explicit formulas for the geodesic curvature and torsion of an arbitrary parameterized curve are obtained using a generalization of vector algebra known as the exterior algebra of multivectors, which we develop in detail. As an example of the application of these results we compute the local properties of the image curve of a spherical motion defined in terms of its instantaneous invariants and canonical coordinate system.