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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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 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. – Will Jagy Nov 4 at 1:52
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 A spherical egg fits into the cone as well, though – Pietro Majer Nov 4 at 21:18

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