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

I establish 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 that the 3D motion of the curve preserves arc-length, a more general assumption than that of rigid motion. I show that, contrarily to what is commonly believed, the full optical flow of the curve can never be recovered from this surface. I nonetheless 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.