How to find the equation of Cardioid using complex numbers? Wikipedia article for cardioid gives a brief proof of parametric Equation Of Cardioid using Complex numbers for representing rotations as following :
Assuming one circle is fixed at
$(-a,0)$ and the rotating one starts from $(a,0)$ . It is given
Rotation around point $a$
$z \rightarrow a+(z-a)e^{i\phi}$
Rotation around point $-a$
$z \rightarrow -a+(z+a)e^{i\phi}$
Then composing one after another taking $z=0$ as starting point.

Can anyone clarify a bit more, where this $a+(z-a)e^{i\phi}$ came from ?  I just know rotating by angle $\phi$ is same as multiplying $e^{i\phi}$, but looking at picture of cardioid, it's not a pure rotation, there is some scaling involved depending upon where you are !
Any kind of help is highly appreciated.
 A: The Wikipedia page is trying to say that a point on the cardioid can be expressed as two successive "shift, rotation, and shift back" sequences, operating on the origin, $z = 0$, as the starting point, without the need for a scaling operation.
Breaking things down:

*

*$(z-a)$ shifts the entire complex plane to be centered at $a$

*multiplication by $e^{i\phi}$ rotates the complex plane about the new origin by $\phi$ radians

*$a + \dots$ shifts the entire complex plane back to the original origin.

*$(z-[-a])$ shifts the entire complex plane to be centered at $-a$

*multiplication by $e^{i\phi}$ rotates the complex plane about the new origin by $\phi$ radians

*$-a + \dots$ shifts the entire complex plane back to the original origin.

A: $$ze^{i\phi}$$ rotates $z$ around the origin, while
$$(z-c)e^{i\phi}+c$$ rotates $z$ around $c$ (you translate $c$ to the origin, rotate and translate back.)
The two rotations account for the fact that the rolling circle rotates around the fixed circle at twice the angular speed of its center.
