Tangent developable

The tangent developable of a helix

The tangent developable of a space curve \gamma(t) is a developable surface formed by the union of the tangent lines to the curve.

Parameterization

Let \gamma(t) be a parameterization of a smooth space curve. That is, \gamma is a twice-differentiable function with nowhere-vanishing derivative that maps its argument t (a real number) to a point in space; the curve is the image of \gamma. Then a two-dimensional surface, the tangent developable of \gamma, may be parameterized by the map

(s,t)\mapsto \gamma(t) + s\gamma'(t).[1]

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.[2] However, not every ruled surface is developable; the helicoid provides a counterexample.

History

Tangent developables were first studied by Leonhard Euler in 1772.[3] 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.[2]

References

  1. ↑ Pressley, Andrew (2010), Elementary Differential Geometry, Springer, p. 129, ISBN 1-84882-890-X.
  2. ↑ 2.0 2.1 Lawrence, Snežana (2011), "Developable surfaces: their history and application", Nexus Network Journal 13 (3): 701–714, doi:10.1007/s00004-011-0087-z.
  3. ↑ Euler, L. (1772), "De solidis quorum superficiem in planum explicare licet", Novi Commentarii academiae scientiarum Petropolitanae (in Latin) 16: 3–34.

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