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