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developable surface

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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 types:

Cylindrical surfaces
Conical surfaces
Tangential surfaces of a space curve; they can be expressed by
$\displaystyle \vec{r} = \vec{\gamma}(t)+ s \frac{d\vec{\gamma}(t)}{dt}$
where $\vec{r} = \vec{\gamma}(t)$ is the equation of the space curve, and are parameters.

**Figure:** Tangential surface of a circular helix
$\includegraphics{developable.eps}$

"developable surface" is owned by pahio. [ full author list (2) ]

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Other names:

torsal surface

Also defines:

developable, torsal generatrix, torsal, tangential surface

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Cross-references: parameters, equation, space curve, conical surfaces, cylindrical surfaces, divide, Gauss, plane, iff, surface, tangent plane, points, ruled surface, generatrix
There are 3 references to this entry.

This is version 5 of developable surface, born on 2005-08-28, modified 2007-04-21.
Object id is 7348, canonical name is DevelopableSurface.
Accessed 4892 times total.

Classification:

AMS MSC:	51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
	51M20 (Geometry :: Real and complex geometry :: Polyhedra and polytopes; regular figures, division of spaces)

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