On the Scalar and Dual Formulations of the Curvature Theory of Line Trajectories

Abstract

The curvature theory of ruled surfaces has been studied in two different ways. The scalar formulation proceeds by defining a seqeunce of ruled surfaces associated with the trajectory ruled surface. The relative positions of these surfaces and their distribution parameters characterize the local properties of the original ruled surface. The other formulation uses dual vector algebra to transform the differential geometry of ruled surfaces into that of spherical curves. In each theory functions are obtained which characterize the shape of the ruled surface. This paper unites these formulations by deriving formulas relating the scalar and dual curvature functions. This provides the ability to compute either set of curvature properties from either the scalar or dual vector representation of the ruled surface. The ruled surface generated by a line fixed in a body undergoing a screw displacement is examined in detail.