From: "Clive Tooth" <pisquaredoversix@pisquaredoversix.force9.co.uk>
Subject: Re: Shortest Distances on Analytical Surfaces (Cylinder, ..)
Date: Sun, 26 Sep 1999 13:27:58 +0100
Newsgroups: [missing]
To: "Dave Rusin" <rusin@math.niu.edu>
Keywords: What are ruled, developable surfaces

Hi Dave!

>In article <7sa4e5$phe$1@mail.pl.unisys.com> you write:
>>Not all ruled surfaces can be "unwrapped".
>
>It's been very long since I studied developpable surfaces. Can you give
>the undergraduates' summary of what theorem/examples you're referring to?
>dave


A ruled surface is the line system generated by a variable line whose
position depends effectively on the value of a single parameter.
http://www.treasure-troves.com/math/RuledSurface.html

In general, a ruled surface cannot be "unwrapped".
Examples are the hyperbolic paraboloid, the helicoid and the cone.

A developable surface is a ruled surface with the property that
"consecutive" generators always intersect. A fancier definition is: any
surface whose gaussian curvature is everywhere zero.
http://www.treasure-troves.com/math/DevelopableSurface.html

Any developable surface can be "unwrapped", ie, unrolled into a plane
without tearing, stretching, creasing, etc.
Of the three ruled surfaces mentioned above only the cone is developable.

Clive Tooth
http://www.pisquaredoversix.force9.co.uk/