# How is a cardioid a special type of limacon?

A limaçon is a curve that is described by the polar equation $r \left(\theta\right) = b + a \cos \left(\theta\right)$.
The cardioid generated by a circle of radius $c$ is the curve described by the polar equation $r \left(\theta\right) = 2 c \left[1 + \cos \left(\theta\right)\right]$.
Defining $d = 2 c$, we get, for the cardioid: $r \left(\theta\right) = d \left[1 + \cos \left(\theta\right)\right]$.
Expanding the previous expression, we get $r \left(\theta\right) = d + d \cos \left(\theta\right)$, and it becomes apparent that the cardioid is the special type of limaçon such that the parameters $a$ and $b$ are equal.