I'm working on coming up with a smooth approximation of the floor function that doesn't use a Fourier series or a step function. Having exhausted more direct methods, I'm looking to see if parametrizing the equation might help so came up with the following parametric equations for Floor(x):
$$ x(t)=\frac{1}{2}\times\left(t+\frac{i}{\pi}\times\ln\left(\sqrt{2a-\cos(2 \pi t)+1}+i\sqrt{2}\cos(\pi t)\right)\right) $$ $$ y(t)=\frac{1}{2}\times\left(t-\frac{i}{\pi}\times\ln\left(\sqrt{2a-\cos(2 \pi t)+1}+i\sqrt{2}\cos(\pi t)\right)\right) $$
These equations should converge to Floor($x$) as $a$ goes to zero. I'm trying to solve $x(t)$ for $t$ so that I can plug it into $y(t)$ for a given $x$. Any ideas if this is solvable?
If it helps, the expression employing the natural log, $\ln\left(\sqrt{2a-\cos(2 \pi t)+1}+i\sqrt{2}\cos(\pi t)\right)$, is derived from the smooth approximation of the triangle wave function.
Thanks!