How to find the equation of Cardioid using complex numbers?

Question

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.

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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.

Answer 1

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:

1. $(z-a)$ shifts the entire complex plane to be centered at $a$
2. multiplication by $e^{i\phi}$ rotates the complex plane about the new origin by $\phi$ radians
3. $a + \dots$ shifts the entire complex plane back to the original origin.
4. $(z-[-a])$ shifts the entire complex plane to be centered at $-a$
5. multiplication by $e^{i\phi}$ rotates the complex plane about the new origin by $\phi$ radians
6. $-a + \dots$ shifts the entire complex plane back to the original origin.

Answer 2

$$ze^{i\phi}$$ 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.