Drawing a Cycloid as an Envelope

Given a family of curves, depending on a continuous parameter, a curve which is tangent to every member of the family is an envelope of the family. Any smooth curve is the envelope of its tangent lines. If the tangent lines are easy to construct, then here is a method for producing a pleasing drawing of the curve.

So it is with the cycloids. A tangent of one of these curves passes through two points P′ and Q′ on the apsidal circle. Here, P′ is the apse of the rolling circle; it is represented by the complex number (1-2w)Re. Similarly, Q′ is the apse of the alternate rolling circle, and its numerical representation is –(1–2w)Rei(1-1/w)θ.

This display draws cycloids as envelopes. The controls which determine the wheel ratio are familiar. Another control determines how many lines are drawn for each arc of the curve. A check box controls whether the cycloid itself is drawn.

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

Christopher J. Henrich