On the motion of 3D curves and its relationship to optimal flow

I establish the fundamental equations that relate the three dimensional motion of a curve to its observed image motion. I introduce the notion of spatio-temporal surface and study its differential properties up to the second order. In order to do this, I only make the assumption than that of rigid motion. I show that, contrarily to what is commonly believed, the full optical flow of the curve (i.e. the component tangent to the curve) can never be recovered from this surface. I also give the equations that characterize the spatio-temporal surface completely up to a rigid transformation. Those equations are the expressions of the first and second fundamental forms and the Gauss and Codazzi-Mainardi equations. I then show that the hypothesis of a rigid 3D motion allows in general to recover the structure and the motion of the curve, in fact without explicitely computing the tangential optical flow, at the cost of introducing the three-dimensional accelerations