Until now in this book we have treated the involute helicoid as a surface generated by a moving line, the line remaining always tangential to a given helix. We have used for the radius of the helix the symbol a, and for the angle of the helix (the helix angle) the symbol α. See figure 3.06. Thus (a, α) has characterized the involute helicoid. We have argued incidentally, not only in general that the pitch p of the involute helicoid is p = a cot α, but also in the special context of gear teeth that the pitch p FLANK of the flank of a tooth is p FLANK = a cot α. Early in chapter 9A we studied the various sets of lines and curves to be found upon the surface of the helicoid. They were (a) the already mentioned straight generating lines, (b) the planar involutes occupying planes drawn normal to the axis of the helicoid, (c) the helices drawn upon circular cylinders coaxial with axis of the helicoid, and (d) the slip tracks. In this chapter 9B, we deal now with other geometrical aspects of the involute helicoid that are important not only in general but also in that they relate to the mechanical processes of the machine-cutting of teeth.
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