I've seen many questions about how to space points out equally on an ellipse, but nothing like my question:
I'm reading a paper using the elliptic coordinates $$x=c\cosh(\xi)\cos(\eta) \\ y = c\sinh(\xi)\sin(\eta) $$
that claims that placing points equally spaced in $\eta$ will result in points being concentrated towards the parts of the ellipse with greatest curvature.
I know that if we look at an ellipse as a stretched circle, the $\eta$ coordinate is like $\theta$ on a circle. Specifically, it's like a circle stretched by factors of $\cosh(\xi)$ and $\sinh(\xi)$ in the $x$ and $y$ directions, respectively.
Is the following thinking correct? : I'm picturing points equally spaced in $\eta$ (i.e. $\theta$) on a circle of radius $c$. Then I'm picturing the stretching map that I just described being applied to each point on the circle. Because $\cosh(\xi) > \sinh(\xi)$ for all $\xi$, the points on the circle will be disproportionately stretched toward the more curved ends of the ellipse, thus concentrating points where the curvature is the greatest.