Definition:

The members of Limacon Family may be generated in the manner of one circle rolling upon another without slipping.

As the figure shown, let the original position of B’ be B, then , where T is the point of tangency, and accordingly ∠ACB＝∠CAB'＝θ. Take the origin of coordinates at O, a distance b from C on the line CB. Dropping perpendicular from O and P upon AC, it is clear that r ＝2a－2bcosθ is the polar equation of the path of P. The three types of this family are defined when

b<a  (p interior to the rolling circle)

b＝a  (p on the rolling circle)

b>a  (p extension and attached to an extension of a diameter)

Theorem:

If , , , then ∠ACK＝∠CAB’.

Pf:

Let ∠ACK＝θ＝∠FCE

∴∠FDE＝θ

∵FEDC∼EDB’A

∴∠EAB’ ＝θ

Use for linkage:

If we fixed the point F and C to the plane, then any point P' of the line AB describes a limacon.

 Furthermore, if AC=AP', then P' describes a Cardiod.

A very similar linkage is given by Hebbert. Again two similar crossed parallelograms FCDK and OGED, are taken to produce equal angle θ at the fixed angle.

Robert C. Yate Geometrical Tools p.182~183 Michigan 1949