As a heads-up: I have no background in mathematics, I am just toying around a bit; I hope that is acceptable.
I am trying to find an algebraic curve that matches a specific shape as good as possible. In detail I want to have the same shape as a normal 400m running track. I am just referring to the red part on which one's running.
I have already come up with the (bad) approximation
$$\left|y\right|\approx 0.00007450531301\sqrt{237510513310-x^{6}}.$$
A perfect solution would be
$$\left|y\right|=36.5\textrm{ with }x\in\left[-42.195;42.195\right]$$
$$\land$$
$$\left(x\pm 42.195\right)^2+y^2=1332.25\textrm{ with }\left|x\right|\in\left[42.195;78.695\right]$$
but I want to work with only one equation.
So I want to approximate a 400m track where one unit of length in the coordinate system equates to 1m in real-life. Here are the standard measurements.
Edit: I think an equation of the form
$$\frac{1}{\sqrt{\left(x-a\right)^{2}+by^{2}}}+\frac{1}{\sqrt{\left(x+a\right)^{2}+by^{2}}}=c$$
can be a very good approximation, but I don't know which values for $a,b,c$ are optimal.
2026-04-24 17:19:39.1777051179
