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The tangent developable of a space curve
Contents
Parameterization
Let
The original curve forms a boundary of the tangent developable, and is called its directrix.
Properties
The tangent developable is a developable surface; that is, it is a surface with zero Gaussian curvature. It is one of three fundamental types of developable surface; the other two are the generalized cones (the surface traced out by a one-dimensional family of lines through a fixed point), and the cylinders (surfaces traced out by a one-dimensional family of parallel lines). (The plane is sometimes given as a fourth type, or may be seen as a special case of either of these two types.) Every developable surface in three-dimensional space may be formed by gluing together pieces of these three types; it follows from this that every developable surface is a ruled surface, a union of a one-dimensional family of lines. However, not every ruled surface is developable; the helicoid provides a counterexample.
History
Tangent developables were first studied by Leonhard Euler in 1772. Until that time, the only known developable surfaces were the generalized cones and the cylinders. Euler showed that tangent developables are developable and that every developable surface is of one of these types.