by Zevan Rosser
In this short article we'll investigate a parametric equation that resembles a ball of yarn. This equation is close to the parametric equation of a circle and directly related to polar coordinates. The cartesian form is:
This results in a network of lines connecting points on the circumference of a circle.
The parametric equation of a circle is:
Where $r$ is the radius and $\theta$ is the parameter in the range $[0,2\pi]$. Having a value of $r=1$ is the same as not having $r$ at all, so we can remove this from the equation. Getting the sine or cosine of large values results in numbers that can appear somewhat random. By cubing $\theta$ we jump to seemingly random points on the circumference of a circle, instead of slowly progressing clockwise from $0$ to $2\pi$ radians.
Altering the exponents of $\theta$ so that they're different for the $x$ and $y$ components renders a rectangular formation:
In the above plot the range for $\theta$ is offset by some value value at least a bit larger than $2\pi$:
Without this offset a slight curve pattern is noticeable on the right hand side of the resulting rectangle. This plot is interesting as it highlights the random nature of large $\theta$ values for sine and cosine.