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):
           Egg Fig
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!
           Egg Fig

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    $\begingroup$ In the ballpark: my father asked once what would happen if you took a bowling ball but installed a steel weight off center, so it was still a sphere outside but the center of mass was off. And then rolled it down a hill. Evidently my brother and I missed the point, as neither of us guessed that it could jump off the hill if rolling fast enough. $\endgroup$ – Will Jagy Nov 4 '12 at 1:52
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    $\begingroup$ A spherical egg fits into the cone as well, though $\endgroup$ – Pietro Majer Nov 4 '12 at 21:18
  • $\begingroup$ The biological premise is charming, but does not cover those birds that lay eggs in a nest, that keep them stable (they do not try and rescue fallen eggs). However, birds that lay eggs on the ground, sometimes even relocate the nest, moving the eggs by making them roll. In this case the egg-shape can actually prevent eggs from rolling down a hill. $\endgroup$ – Pietro Majer Mar 19 '18 at 19:46

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