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I'm an artist trying to sculpt shapes that I would describe as ellipsoids. I am wanting to sew together fabric to form these shapes, and although I know the dimensions of the ellipsoid I want to form, I don't know how to calculate the curvature of the pieces of fabric so they come together to form the ellipsoid (similar to gores on a globe).

For a sphere, I found the formula for the edges of the fabric to be: y = +/- R (pi/N) cos(x/R). N is the amount of gores, and x is the angular height. Source: https://www.themathdoctors.org/making-a-sphere-from-flat-material/?unapproved=21662&moderation-hash=f9644b82fc5759e707821810bd9753e8#comment-21662

But I'm not sure how to translate this formula into elliptical terms. The ellipsoid I'm trying to form has radii of 4.5 and 2.75 in. Does anyone have any suggestions, or know where I could turn to study this more?

kris
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Interesting question. I assume your gores are lens shaped and flat. You make $N$ large enough so that the completed figure is spherical enough for your purpose.

Now note that your ellipsoid is just a sphere stretched out uniformly along the $x$ axis. So you can just stretch that dimension of the gores.

In this image

enter image description here

with $N$ gores the gore when flattened has central width $$ w = \frac{a \pi}{N} . $$ Its edges have equation $$ y = \pm w \cos(\pi x/b) . $$

Ethan Bolker
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  • thanks! I'll give that a try, though I really am looking for an equation for the curve itself... I'm looking at the formulas at https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid ...and hoping something under "equations for a geodesic" might apply? – kris Mar 12 '23 at 02:29
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    No, you don’t want to get mixed up with geodesics. You get the ellipsoid equations just by “stretching” the sphere equations. – bubba Mar 12 '23 at 07:51
  • @EthanBolker thank you SO much, I can't tell you how much this helps! And seeing it drawn out is so nice of you. I really thought it would be so much more complicated! – kris Mar 12 '23 at 20:21
  • @bubba thanks for clarifying that - is it because this is at a smaller scale that geodesics are unnecessary? Or is this actually the most accurate way to calculate any segment of an ellipsoid? – kris Mar 12 '23 at 20:25
  • I doubt that geodesics are especially good splitting curves, and they are very complex, so not worth the trouble. With suitable stretching, the splitting curves you’re using for spheres will work just as well (or better) on ellipsoids. – bubba Mar 12 '23 at 23:39
  • @bubba thank you – kris Mar 13 '23 at 02:32