Modern practice has accepted the curve known in general terms as the cycloid, as that best filling all the requirements of wheel teeth, and this curve is employed to produce two distinct forms of teeth, epicycloidal and involute. In epicycloidal teeth the curve forming the face of the tooth is designated an epicycloid, and that forming the flank an hypocycloid. An epicycloid may be traced or generated, as it is termed, by a point in the circumference of a circle that rolls without slip upon the circumference of another circle. Thus, in Fig. 18, A and B represent two wooden wheels, A having a pencil at P, to serve as a tracing or marking point. Now, if the wheels are laid upon a sheet of paper and while holding B in a fixed position, roll A in contact with B and let the tracing point touch the paper, the point P will trace the curve C C. Suppose now the diameter of the base circle B to be infinitely large, a portion of its circumference may be represented by a straight line, and the curve traced by a point on the circumference of the generating circle as it rolls along the base line B is termed a cycloid. Thus, in Fig. 19, B is the base line, A the rolling wheel or generating circle, and C C the cycloidal curve traced or marked by the point D when A is rolled along B. If now we suppose the base line B to represent the pitch line of a rack, it will be obvious that part of the cycloid at one end is suitable for the face on one side of the tooth, and a part at the other end is suitable for the face of the other side of the tooth.
A hypocycloid is a curve traced or generated by a point on the circumference of a circle rolling within and in contact (without slip) with another circle. Thus, in Fig. 20, A represents a wheel in contact with the internal circumference of B, and a point on its circumference will trace the two curves, C C, both curves starting from the same point, the upper having been traced by rolling the generating circle or wheel A in one direction and the lower curve by rolling it in the opposite direction.
To demonstrate that by the epicycloidal and hypocycloidal curves, forming the faces and flanks of what are known as epicycloidal teeth, motion may be communicated from one wheel to another with as much uniformity as by frictional contact of their circumferential surfaces, let A, B, in Fig. 21, represent two plain wheel disks at liberty to revolve about their fixed centres, and let C C represent a margin of stiff white paper attached to the face of B so as to revolve with it. Now suppose that A and B are in close contact at their perimeters at the point G, and that there is no slip, and that rotary motion commenced when the point E (where as tracing point a pencil is attached), in conjunction with the point F, formed the point of contact of the two wheels, and continued until the points E and F had arrived at their respective positions as shown in the figure; the pencil at E will have traced upon the margin of white paper the portion of an epicycloid denoted by the curve E F; and as the movement of the two wheels A, B, took place by reason of the contact of their circumferences, it is evident that the length of the arc E G must be equal to that of the arc G F, and that the motion of A (supposing it to be the driver) would be communicated uniformly to B.
