Okay, i had a very strange thought, it was
"Is it possible to find the length of an elliptical spiral whose major and minor axes were decreasing?"
Like for example lets say that
$$ \frac{a}{b} = n $$
Then if the major axis's decrease can me shown as $\frac{a}{k^2+1}$ and the minor axis's decrease can be shown as $\frac{b}{k^2+n}$ then
$$ e = \sqrt{1-\frac{k^2+n}{k^2+1}} = \sqrt{\frac{1-n}{k^2+1}} $$
Therefore is i define the spiral length as
$$ S = a \sum_{k=0}^{\infty} \frac{E(\sqrt{\frac{1-n}{k^2+1}})}{k^2+1} $$
Where $E(\sqrt{\frac{1-n}{k^2+1}})$ is the complete elliptic integral of the 2nd kind. This is defined as
$$ \int_0^{\pi/2}\sqrt{1-\frac{\sin^2(\Theta)(1-n)}{k^2+1}}d\Theta $$
The series converges, But im unsure of how to come up with the values. I don't even know how to try.