How is a cardioid a special type of limacon?

1 Answer

A limaçon is a curve that is described by the polar equation #r(theta)=b + a cos(theta)#.

The cardioid generated by a circle of radius #c# is the curve described by the polar equation #r(theta)=2c [1+cos(theta)]#.

Defining #d=2c#, we get, for the cardioid: #r(theta)=d[1+cos(theta)]#.

Expanding the previous expression, we get #r(theta)=d+d cos(theta)#, and it becomes apparent that the cardioid is the special type of limaçon such that the parameters #a# and #b# are equal.