Bridging the gap between rectifying developables and tangent developables: a family of developable surfaces associated with a space curve

Abstract

There are two familiar constructions of a developable surface from a space curve. The tangent developable is a ruled surface for which the rulings are tangent to the curve at each point and relative to this surface the absolute value of the geodesic curvature κ g of the curve equals the curvature κ . The alternative construction is the rectifying developable. The geodesic curvature of the curve relative to any such surface vanishes. We show that there is a family of developable surfaces that can be generated from a curve, one surface for each function k that is defined on the curve and satisfies | k | ≤ κ , and that the geodesic curvature of the curve relative to each such constructed surface satisfies κ g = k .

Nonrigidity of flat ribbons

“…In fact, a closely related work [22] appeared shortly before the first version of this paper was completed. By basing their analysis on the geodesic curvature – rather than the ruling angle – the authors in [22] offer an alternative description of the surfaces studied here.…”

Caustics of developable surfaces

“…Two of them are the well-known cone and cylinder, while the third one is a more general type, namely, the tangent surface of spatial curves. From the computational point of view, this latter type is the most challenging one in engineering applications, but at the same time, this type provides much more freedom in engineering design than the classical cones and cylinders (Seguin et al, 2021). These surfaces are used, among other applications, for creating special developable mechanisms, e.g., for cylindrical (Greenwood et al, 2019) and for conical (Hyatt et al, 2020).…”