From: "Clive Tooth" Subject: Re: Shortest Distances on Analytical Surfaces (Cylinder, ..) Date: Sun, 26 Sep 1999 13:27:58 +0100 Newsgroups: [missing] To: "Dave Rusin" 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/