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It is known that any ellipsoid with principal semi-axes $a$, $b$, $c$ has circular planar sections (https://en.wikipedia.org/wiki/Circular_section).

  • Is the largest circular disk contained within any ellipsoid given by one of its circular sections?

  • What is the expression for the radius and other parameters of the largest contained disk in an ellipsoid in terms of $a$, $b$, $c$?

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  • $\begingroup$ Doesn’t the linked wiki article already give a complete answer? $\endgroup$
    – Pietro Majer
    Commented Aug 3, 2021 at 14:11
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    $\begingroup$ Thanks. I could not figure out the answer. The largest disk inside an ellipsoid could well be a subset of an elliptical cross section of the ellipsoid - I am not sure if such a possibility is ruled out. $\endgroup$
    – Nandakumar R
    Commented Aug 3, 2021 at 14:28
  • $\begingroup$ As per the linked wiki article, any circle in the ellipsoid $x^2/a^2+y^2/b^2+z^2/c^2=1$ with $a<b<c$ belongs to a family of circles obtained as sections with a family of parallel planes. The extremal planes are tangent planes to umbilical points. $\endgroup$
    – Pietro Majer
    Commented Aug 3, 2021 at 14:50
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    $\begingroup$ So in this case the question is equivalent to: any plane section of the ellipsoid is an ellipse whose minor semi-axis is $\le b$. $\endgroup$
    – Pietro Majer
    Commented Aug 3, 2021 at 15:00
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    $\begingroup$ Thanks very much. I guess the last comment answers the question - the radius of the largest disc contained in an ellipse is given by the second largest of the 3 semi axes. $\endgroup$
    – Nandakumar R
    Commented Aug 3, 2021 at 15:15

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