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 curve; they can be expressed by

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

where  $\vec{r}=\vec{\gamma}(t)$  is the equation of the space curve, $s$ and $t$ are parameters.

 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