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 κg 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 κg = k.
Highlights
The two fundamental notions of curvature for a surface in three space dimensions are the mean and Gaussian curvatures
The purpose of deriving these restrictions is to motivative the assumptions underlying the main result established in §3, which states that given a curve C and a function k satisfying certain conditions which we delineate, it is possible to construct a developable surface S such that the geodesic curvature of C relative to S is k
The main idea underpinning the theorem that we state and prove below is that, given a curve C and a function k satisfying certain conditions, it is possible to generate a ruled surface S with vanishing Gaussian curvature K such that C lies on S and has geodesic curvature κg equal to k
Summary
The two fundamental notions of curvature for a surface in three space dimensions are the mean and Gaussian curvatures. To allow for situations in which the curvature κ of C vanishes on a finite number of intervals, we stipulate that k complies with a jump-like condition consistent with changes in orientation, about the tangent to C, of the Frenet frame at junctions between curved and straight segments of C. This is condition (C2) in theorem 3.1. The purpose of deriving these restrictions is to motivative the assumptions underlying the main result established in §3, which states that given a curve C and a function k satisfying certain conditions which we delineate, it is possible to construct a developable surface S such that the geodesic curvature of C relative to S is k
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