Inextensible flows of curves and developable surfaces

The flow of a curve or surface is said to be inextensible if, in the former case, the arclength is preserved, and in the latter case, if the intrinsic curvature is preserved. Physically, inextensible curve and surface flows are characterized by the absence of any strain energy induced from the motion. In this paper we investigate inextensible flows of curves and developable surfaces in R3. Necessary and sufficient conditions for an inextensible curve flow are first expressed as a partial differential equation involving the curvature and torsion. We then derive the corresponding equations for the inextensible flow of a developable surface, and show that it suffices to describe its evolution in terms of two inextensible curve flows.