In Fig. 4 of the following paper, http://ruina.tam.cornell.edu/research/topics/locomotion_and_robotics/papers/... , an "oval" is show to roll and hop along a horizontal line without losing any energy. The center-of-gravity (cog) of the oval is a periodic sequence of upward-facing circular arcs and downward facing parabolic arcs. The speed of travel along the x-coordinate is synchronized with the rotation of the oval, so that the oval touches down in a way entirely symmetrical with the way it lifts off. Furthermore, only one side of the oval ever touches the ground, as the other side is up in the air when the oval revolves before touching down again (I assume that this situation could be generalized to do more than 1/2 rotation in the air). The question: what is the mathematical shape of the oval to achieve this overall lossless rolling & hopping behavior? (I don't know the answer, and the paper didn't give any references that I could download from the Internet.) Additional questions might be: does the oval have to be symmetrical about its short axis? (Maybe not.) Does the cog have to traverse a circular arc during the actual rolling motion? (I think not, so long as the velocity & angular velocity on point of contact and velocity & angular velocity are the "same" (modulo mirror images).) Has anyone else run across this type of oval before? Apparently, a football rolling end-over-end can approximate some of this behavior. I'm surprised that no one has built a children's toy based on this principle -- a hopping hoop would be much more fun than a simple rolling hoop.