Geometric interpretation of the elliptic integrals

Question

The elliptic integrals seem to be constructed in a similar manner to the trigonometric and hyperbolic ones. We take a shape, in the elliptic case, an ellipse. $$\Delta= \sqrt{(1-x^2)(1-k^2x^2)}$$ Then we invert and put in an integral. $$\int \frac{1}{\Delta}$$ And then we get a set of Inverse functions.
What is the geometric interpretation of taking a shape inverting it(in the fraction sense) taking an integral and then finding the resulting function’s inverse?
This might involve the inverse function theorem.
Moreover, is it useful to do this for other shapes? Such as the hyperbolic counterpart of an ellipse?