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Caption: Mathematical Models Catalog of a Collection of Models of Ruled Surfaces
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32 Developable Surfaces. If w e consider a stiff card, of the form a X Y f E in the figure above, to be deeply scored along the right lines B A J , 0 B c9 D C d, E D e, so that w e can bend it along each of them, the broken line A B O D E will form a polygon,, at first plane, but a skew polygon when w e come to bend the the surface, which will then form a polyhedron, every edge of which will run into every successive edge, along the broken line A B O D E . It is evident that this condition is necessary to our being able to deform the surface. For if one of the scores (say c B ) stopped short of the edge A B 0, the card would not bend, and if A B 0 were not an actual edge, in that case it would not bend either. N o w if w e consider a surface which can be formed by the gradual bending of a plane surface, the only departures from this type are,—(1.) That the polyhedral surface is replaced by a curved surface. (2.) Tiiat the polygon A B 0 D E is replaced by a curve. (3.) That instead of a finite bending along a few lines A a, B &, 0 c, &c. w e have an infinitesimal bending along an infinite number of such lines infinitely close together. (4.) That all these lines are tangents to the curve A B C , which replaces the polygon. (5.) That the consecutive lines A «, B h, meet one another; that is to say, the shortest distance between them is an infinitesimal of a higher order than the distance between any other two points of them. This is implied in their being tangents to the same curve.
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