From Parametric to Polar in Desmos
This post was originally a Twitter thread. It started with a @DailyDesmos-style challenge from @CmonMattTHINK. I replied with a slight modification and then with a version after some additional parametric machinations, both shown below. Then @matheser asked how I transformed the curve circlewise, and this thread was my answer.
@matheser @CmonMattTHINK @DailyDesmos Sure! Step 1: I started with my original parametric function (x(t),y(t)), applied a vertical translation to get (x(t), y(t)+h) so that the entire graph lies above the x-axis.
Step 2: Apply a polar transformation (r,theta) -> (r cos(theta), r sin(theta)). In effect, think of the x-coordinate above as the angle and the y-coordinate above as the radius. (That is why you want to do the vertical shift.)
This results in the function ((y(t)+h) cos(x(t)), (y(t)+h) sin(x(t))). Hope this helps!
(Realizing now that many people would write k instead of h. I hope this isn’t a source of confusion.)
Citation
@online{hanusa2021,
author = {Hanusa, Christopher R. H.},
title = {From {Parametric} to {Polar} in {Desmos}},
date = {2021-07-06},
url = {https://hanusa.xyz/blog/posts/2021-07-06-from-parametric-to-polar-in-desmos/},
langid = {en}
}