An Angel Function!

I was playing with Desmos and I needed some function that would do some interesting things around a circle. I was tinkering, and accidentally found a function that draws something like an angel with wings stretched out :)

$x = \left(2+2\ \sin t\right)\cos t$ ,

$y = \left(2+2\ \sin\ 3t\right)\sin\ t$ ,

$0 < t < 2\pi$

It looks like this. Does it look like an angel to you? The shaded and dotted circles are embellishments :)

Play with the function in Desmos: https://www.desmos.com/calculator/xzsealifzf

Analysis

We can un-roll the function on a linear axis to see the critical points. The radius at an angle t then would be:

$y=\sqrt{\left(\left(2+2\sin t\right)\cos t\right)^{2}+\left(\left(2+2\sin3t\right)\sin t\right)^{2}}$

From $0 < t < 2\pi$ , we see there are 6 critical points. Looking at the shape, we would expect 8 of them, but the bends at the lower base of the wings (C and G) are actually not critical points on the radius function. At $(\pi/2, 0)$ is the absolute minima, and $(3\pi/2, 4)$ , the absolute maxima.

Variations

This produces something like an alien!

$x = \left(2+2\ \sin t\right)\cos t$ ,

$y = \left(2+2\ \sin\ 1.4t\right)\sin\ t$ ,

$0 < t < 10\pi$

Ice cream cone?

$x = \left(2+2\ \sin t\right)\cos t$ ,

$y = \left(2+2\ \sin\ 5.7t\right)\sin\ t$ ,

$0 < t < 30\pi$

My friend Mani observes that for $y = \left(2+2\ \sin 7t\right)\sin\ t$ , it looks like a strange sea creature we see in National Geographic Channel! I stretched it out on X-axis a bit, added some eyes and mouth, and it looks like this: