I am seeking a mathematical analysis of an egg-ovoid rolling down an inclined plane, for pedagogical reasons. It is well-known folk lore that the shape of an egg prevents it from rolling away from the mother's nest, e.g., the article "Why are eggs egg-shaped?" (link), or, a bit more formally, "The Mathematics of Egg Shape" (PDF download) by Yutaka Nishiyama, from which I copied this figure (and the one below):
           
If anyone has seen a mathematical analysis of this phenomenon, more precise than observing that an egg-ovoid fits inside a cone, and a cone rolls a circular arc, I would appreciate a pointer. Thanks!