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Suppose you have an ellipsoid given by the set,

$$\left\{ x \in\mathbb{R}^3 \mid x^TQx = 1 \right\}$$

where $Q = \mbox{diag}(a,b,c)$. Is there a way to parametrize the set

$$\left\{ x \in \mathbb{R}^3 \mid \|p-x\| = k, x^TQx = 1 \right\}$$

where $Q = \mbox{diag}(a,b,c)$, $p\in\mathbb{R}^3$, and $k$ is a real number. In the case I care about, $a=b$, but I am unsure if that simplifies the problem much, and the general case might be helpful to others. Notice if $a=b=c$ the set is a circle and it is relatively easy to parametrize.

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  • $\begingroup$ If $\|p\|$ is large, the set you're hoping to parameterize may well be empty, so you have to expect at least a few "if" clauses in any solution. $\endgroup$ Commented Jul 21, 2020 at 12:45
  • $\begingroup$ @Hughes I am interested in the case where the intersection is a closed curve. $\endgroup$
    – math314
    Commented Jul 21, 2020 at 13:32
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    $\begingroup$ I don't think that there is a nice solution. You could express all points $(x,y,z)$ on the sphere as a function of two variables $u$ and $v$ by means of a stereographic projection. Then use $u$ as the parameter and solve $ax^2+by^2+cz^2=1$ for $v,$ which will require to solve a quartic equation. $\endgroup$ Commented Jul 21, 2020 at 14:13
  • $\begingroup$ @Reinhard Meier I agree with you ; this quartic equation will probably be factorizable in the special case where $p$ is situated so as to generate 2 intersection curves (for example when $p$ is at the origin). $\endgroup$
    – Jean Marie
    Commented Jul 23, 2020 at 7:19

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