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This is to complete the conic locus picture following my post "Ellipse like on a sphere", answered by achille hui.

Find the locus of all points on a sphere such that the difference of geodesic distances from two fixed points $F_1$ and $F_2$ on it is a constant, less than its radius. (When radius of sphere goes to infinity, the loci would look like hyperbolae).

Perhaps hui's derivation would proceed along same lines incorporating the minus sign.

This result of " ellipse-like on a sphere " locus is beautiful. In the sense anyone who knew how to trace ellipse locus between two pegs and taut string would perhaps want to formulate to be able to see a generalization on a sphere. I.e., we seek loci for constant sum & difference of focal distances generalized /transferred from flat plane on a sphere.

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Narasimham
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  • Eq. $(*1)$ continues to work except $D = \tan\theta$ is now smaller than $A$.
  • When you project the loci to the $xy$-plane, they are now a family of hyperbolas.
  • If you project the loci the $yz$-plane, they are a family of ellipses. This is

    • because $$PF_1 - PF_2 = 2\theta;;\iff;; (\pi - PF_1') - PF_2 = 2\theta ;;\iff;; PF_1' + PF_2 = \pi - 2\theta$$ where $F_1'$ is the antipodal point of $F$ on the sphere.

    – achille hui Aug 25 '14 at 07:29