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


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.


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


2 Replies to “An Angel Function!

  1. 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.

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