Bridging the gap between rectifying developables and tangent developables: a family of developable surfaces...

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 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  = k.