While reading up on Limaçons I learned they are the envelope of one circle around another.
The limaçon can be generated by specifying a fixed point P, then drawing a sequences of circles with centers on a given circle C which all pass through P. The envelope of these curves is a limaçon. If the fixed point is on the circumference of the circle, then the envelope is a cardioid.
In fact, I even drew the foliation myself, exploring how the circles change as you move the fixed point P.
I gather that envelopes are often 1-parameter families of solutions to differential equations. This is related to an old-fashioned way of defining Lie Groups as symmetries of differential equations.
We can directly compute the function $f:(P, \theta)\mapsto (x,y,r)$ taking each point $P$ in the plane and angle $\theta$ (relative to the vertical) on the circle C.
What is the map here? What is that differential equation here?
Huygens principle may be involved here, so can this foliation be studied with contact geometry?