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^{iθ}.
Similarly, Q′ is the apse of the alternate rolling circle, and its numerical
representation is –(1–2w)Re^{i(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.