Is it possible to construct functions analogous to sine and cosine, or in the case of elliptical functions, sn, cn, etc., for a 3-ellipse? Similarly,would $\pi$ play any role in calculating the area, or would another constant be needed?
Another question I have is regarding the radius of a 3-ellipse. The few articles I've been able to find all claim that the the radius is defined to be the Geometric Median. What's unclear to me is that there's no clear way to convert the (x,y) coordinates of the median to the smallest possible radius. This also makes me wonder is it possible to way to find the smallest radius, without computing the median?
I was able to find the following formula to find the minimal radius for a 3-ellipse, but no derivation was given, and it's unclear if it can be generalized to a k-ellipse.
$h=a^{2}+b^{2}+c^{2}$
$g=a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}$
$R=\frac{1}{\sqrt{2}}\sqrt{\sqrt{12g-3h^{2}}+h}$
Where a, b and c are the side lengths of the triangle formed by the three foci.
I've created a desmos page that includes a 3-ellipse and also, the analytical solution for the calculating the median.