Equation of cardioid Can someone please explain to how
$$c = \frac{1}{2} e^{i \theta}- \frac{1}{4} e^{2 i \theta}, \quad 0 \leq \theta \leq 2 \pi$$
does represent an equation of cardioid  $r =  \frac{1}{2}  -  \frac{1}{4} \cos(\theta)$, $0\leq \theta \leq 2 \pi$?
I tried to write  $\cos(\theta) = \frac{e^{i \theta} + e^{-i \theta}}{2}$, but couldn't prove it is equal to $\frac{1}{2} e^{i \theta}- \frac{1}{4} e^{2 i \theta}$. Thanks in advance.

For a given value of real $c1$ ( c,s are cos t ,sin t)
the 2D $t$ parametrization is
$$  X(t)= c1 (c/2-(c^2-s^2)/4, (1-c)s/2) $$
For complex $c1= e^ \alpha, $ there is a rotation of the full cardoid by angle $\alpha$.
