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:

x = \left(3+2\ \sin t\right)\cos t ,

y = \left(2+2\ \sin\ 7t\right)\sin\ t ,

0 < t < 2\pi

https://www.desmos.com/calculator/n8sqt6m9dz

## 2 Replies to “An Angel Function!”

1. Mani says:

One suggestion on the article, to make it relatable for those who are not used to the specific equation, you could describe about them in a short, so that, the reader can carry more than just a magic out of it, can understand the relation between the circle , identity, trigonometry and for others it could re-establish the lost interest on subject also.

2. Biddu says:

Nice