84 TRANSFORMATION OF SURFACES BY BENDING. 



Since in this case the quantities , 6, and cr are represented by distinct 

 geometrical quantities, we may simplify the consideration of all surfaces generated 

 by straight lines by reducing them by bending to the case in which those lines 

 are parallel to a given plane. 



In the class of surfaces in which the generating lines ultimately intersect, 



-1^ = 0, and constant. If these surfaces be bent so that <f> = Q, the whole of 



' f '/ 



the generating lines will lie in one plane, and their ultimate intersections will 

 form a plane curve. The surface is thus reduced to one plane, and therefore 

 belongs to the class usually described as "developable surfaces." The form of a 

 developable surface may be defined by means of the three quantities 6, a- and 

 ^. The generating lines form by their ultimate intersections a curve of double 

 curvature to which they are all tangents. This curve has been called tl it- 

 cuspidal edge. The length of this curve is represented by cr, its absolute 



curvature at any point by - , and its torsion at the same point by -7- . 



When the surface is developed, the cuspidal edge becomes a plane curve, 

 and every part of the surface coincides with the plane. But it does not follow 

 that every part of the plane is capable of being bent into the original form 

 of the surface. This may be easily seen by considering the surface when the 

 position of the cuspidal edge nearly coincides with the plane curve but is not 

 confounded with it. It is evident that if from any point in space a tangent 

 can be drawn to the cuspidal edge, a sheet of the surface passes through that 

 point. Hence the number of sheets which pass through one point is the same 

 as the number of tangents to the cuspidal edge which pass through that 

 point ; and since the same is true in the limit, the number of sheets which 

 coincide at any point of the plane is the same as the number of tangents 

 which can be drawn from that point to the plane curve. In constructing a 

 developable surface of paper, we must remove those parts of the sheet from 

 which no real tangents can be drawn, and provide additional sheets where more 

 than one tangent can be drawn. 



In the case of developable surfaces we see the importance of attending to 

 the position of the lines of bending; for though all developable surfaces may 

 be produced from the same plane surface, their distinguishing properties depend 

 on the form of the plane curve which determines the lines of bending. 



