The Evolute of a Cycloid is a Similar Cycloid

If P is a point on a smooth curve C, then among all the lines through P there is one, the tangent T, which has second-order contact with C. Among the circles that are tangent to T at P, the one that has third-order contact with C is called the osculating circle, or circle of curvature, at P. Its radius is the radius of curvature at P, and its center is the center of curvature at P.

The locus of centers of curvature of C is called the evolute of C. The evolute is also the envelope of the straight lines normal to C.

Suppose that C is a trochoid, and suppose a point moving along it with parameter θ increasing at a uniform rate. We have seen that the normal line at the point with parameter θ intersects the supporting circle at the point with angle θ, and it intersects the alternate supporting circle at the point with angle (1-1/w)θ. When the trochoid is in fact a cycloid, then two supporting circles coincide, so the normal is determined by two points which travel around that circle at differing speeds. But we have also seen that, in this case, the tangent line to the same cycloid is determined by two points which travel in a similar way on the apsidal circle. Thus the normal lines to one cycloid are also tangent lines to another one; the supporting circle of the former is the apsidal circle of the latter. The latter is the evolute of the former.

Here is a display of a cycloid, together with its normal lines extended to touch the evolute. The controls are like those of the envelope applet. Under the control of a checkbox, the applet shows or hides an osculating circle, and a knob by which it can be moved to another point.

Last modified on $Date: 2015-05-02 12:51:16 -0400 (Sat, 02 May 2015) $

Christopher J. Henrich