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On the convex part of the surface we place a curve (not necessarily flat, as in this case). We divide this curve into segments of equal length (in the text Ls [i]) and divide the path that our surface will roll (in the text L [i]) into segments of the same length as segments of curve. Take the next segment of the trajectory L [i] and the corresponding segment on the curve Ls [i], calculate the angles between them. After that, we perform well-known transformations that place the curve in the space so that the segment Ls [i] coincides with the segment L [i]. At the same time, we perform exactly the same transformations with the equation of surface.

For example, the ellipsoid rolls on the oX1 axis, and each position of the ellipsoid in space corresponds to the equation in the figure.
Rolling_of_surface.mw

 

And similar examples:
animation surface rolling draghilev

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