Question

How is a cardioid a special type of limacon?

Answer 1

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.