Orbits of balls in a funnel and a spherical surface

Many articles have been published concerning experiments with funnels and other curved surfaces to illustrate Kepler's laws of planetary motion.$^{1-5}$ The main objectives of those experiments are to demonstrate in a qualitative manner that a ball rolling in a funnel follows an orbit that is approximately elliptical and that angular momentum is conserved. It doesn't matter that Kepler's third law is not obeyed, since the demonstrations raise many interesting questions that are not usually described in introductory physics classes.$^{5}$ Nevertheless, the ball orbits that are observed can be described in relatively simple terms, even if they are not measured. Published orbits are difficult to locate in the literature. Consequently, the author set out to measure some orbits in a funnel and in a concave lens. The results are presented below, together with explanations.