I came to this equation while trying to get the time a ball takes in a given surface under the influence of gravity. (Monotonically decreasing height is used to keep it simple)
$$\int_{h_i}^0 \sqrt \frac {h-h_i} {{1+f'(h)^2}} dh $$ Note that $(f(h),h)$ would be the path the particle follows, always staying in contact with the surface.
Is it possible to solve this generally, for unknown f?