Parametric Yarn

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:

$$ x = \cos \theta^3 \\ y = \sin \theta^3 $$

This results in a network of lines connecting points on the circumference of a circle.

The parametric equation of a circle is:

$$ x = r \cos \theta \\ y = r \sin \theta $$

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.

Rectangular Distribution

Altering the exponents of $\theta$ so that they're different for the $x$ and $y$ components renders a rectangular formation:

$$ x = \cos \theta^4 \\ y = \sin \theta^2 $$

In the above plot the range for $\theta$ is offset by some value value at least a bit larger than $2\pi$:

$$ a = 10 \\ \theta = [a, a + 2\pi] $$

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.