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. Cylindrical surfaces

2. Conical surfaces

3. Tangential surfaces of a space curve; they can be expressed by

   $$\overrightarrow{r}=\overrightarrow{\gamma }(t)+s\frac{d\overrightarrow{\gamma }(t)}{dt}$$

   where  $\overrightarrow{r}=\overrightarrow{\gamma }(t)$  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