The full and Essential Condition for a Ruled Surface Being a Developable Surface

The developable surface,along every straight line with only one each tangent plane,is a type of ruled surface.Our purpose is to give a sufficient and necessary condition of developable surface with an enveloping surface.We use the methods of geometry analysis to study ruled surface and get a sufficient and necessary condition of developable surface so that a ruled surface become to a developable surface.Our results are: let S∶r→(u,v)=ρ→(u)+ve→(u) be a ruled surface,we get that S be a developable surface if and only if S be an enveloping surface of tangent plane families along a directrix curve.And it gives two examples of this new application of the theorem.