To obviate the necessity of finding the necessary radius from rolling segments various forms of construction are sometimes employed.
Fig. 132.
Thus Rankine gives that shown in [Fig. 132], which is obtained as follows. Draw the generating circle d, and a d the line of centres. From the point of contact at c, mark on circle d, a point distance from c one-half the amount of the pitch, as at p, and draw the line p c of indefinite length beyond c. Draw a line from p, passing through the line of centres at e, which is equidistant between c and a. Then multiply the length from p to c by the distance from a to d, and divide by the distance between d and e. Take the length and radius so found, and mark it upon p c, as at f, and the latter will be the location of centre for compasses to strike the face curve.
Fig. 133.
Another method of finding the face curve, with compasses, is as follows: In [Fig. 133], let p p represent the pitch circle of the wheel to be marked, and b c the path of the centre of the generating or describing circle as it rolls outside of p p. Let the point b represent the centre of the generating circle when that circle is in contact with the pitch circle at a. Then from b, mark off on b c any number of equidistant points, as d, e, f, g, h, and from a, mark on the pitch circle, points of division, as 1, 2, 3, 4, 5, at the intersection of radial lines from d, e, f, g, and h. With the radius of the generating circle, that is, a b, from b, as a centre, mark the arc i, from d the arc j, from e the arc k, &c., to m, marking as many arcs as there are points of division on b c. With the compasses set to the radius of divisions 1, 2, step off on arc m the five divisions, n, o, s, t, v, and v will be a point in the epicycloidal curves. From point of division 4, step off on l four points of division, as a, b, c, d, and d will be another point in the epicycloidal curve. From point 3 set off three divisions on k, from point 2 two dimensions on l, and so on, and through the points so obtained, draw by hand or with a scroll the curve represented in the cut by curve a v.