developable surface

A generatrix of a ruled surface is torsal, if in each of its points there is one and the same tangent plane of the surface.

A ruled surface is torsal iff it only has torsal generatrices.

A surface is developable, if one can spread it out on a plane without any stretching or tearing.

K. F. Gauss has proved that a surface is developable if and only if it is a torsal ruled surface.

One may divide the developable surfaces into three :

  1. 1.
  2. 2.
  3. 3.

    Tangential surfaces of a space curveMathworldPlanetmath; they can be expressed by


    where  r=γ(t)  is the equation of the space curve, s and t are parameters.

Figure 1: Tangential surface of a circular helix
Title developable surface
Canonical name DevelopableSurface
Date of creation 2013-03-22 15:29:29
Last modified on 2013-03-22 15:29:29
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Topic
Classification msc 51M20
Classification msc 51M04
Synonym torsal surface
Related topic Area2
Related topic RiemannMultipleIntegral
Defines developable
Defines torsal generatrix
Defines torsal
Defines tangential surface