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Given a number of circular beads of a given size strung on an ellipse with a known semi-minor axis, how can I calculate the position (let's say in Cartesian coordinates) of each bead's center?

The semi-major axis is not known, nor is the circumference, really. I initially thought the circumference was the combined diameters of the beads, but I think that would just be an approximation, especially around the apoapsis.

I would include a drawing here, but I don't have enough reputation.

Along the way, I'll probably need to calculate the semi-major axis (a) from the semi-minor (b) and the circumference. Is there an equation I can use, or do I need to solve c = 4 a E(e) by inverting the Elliptic integral?

Dan Eckhart
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You'll need to know more in addition to the bead size and the semi-minor axis:

  • The origin of the ellipse
  • The angle that the major axis makes with the $x$ axis
  • The semi-major axis (or the eccentricity)
  • The location of one of the beads

(I assumed two dimensions; you'll need more if you're working in three.)

From the first bead, you can build up the positions of the rest of the beads at first under the assumption that they touch. They may not meet up at the other side of the ellipse, in which case you'll have a known gap. Then, you can leave them as-is, or distribute the gap evenly.

John
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