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The general equation of a cone that passes through the origin is $$ax^{2}+by^{2}+cz^{2}+2fyz+2gzx+2hxy=0$$If I'm given $5$ points on the cone, I should be able to get $5$ equations and be able to solve for $b,$ $c,$ $f,$ $g,$ and $h$, if I assume the value of $a$ to be, say, $1$.

But if the $5$ points lie on the same generator, I think it wouldn't be possible to solve the system of equations. So I think, the $5$ points should lie on different generators, in order for us to uniquely determine the cone. (Am I correct?)

But what bugs me is the fact that even if this was true, the shape of the cone will be ultimately decided by the shape of the guiding curve or loop in the space. But how can $5$ arbitrary points on the cone determine the shape of the guiding curve, which can practically be of any shape and is the deciding factor in the shape of the cone?

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  • $\begingroup$ Do you agree that the equation is not that of a cone of revolution but a more general one ? $\endgroup$
    – Jean Marie
    Mar 10, 2023 at 13:02
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    $\begingroup$ A cone of revolution is a tangent to a sphere compared for example to a tangent to an ellipsoid. $\endgroup$
    – Jean Marie
    Mar 10, 2023 at 16:14
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    $\begingroup$ @JeanMarie Ok, Thanks. Anyway, I think I understand the issue now. The equation I gave in the question is the equation of a QUADRIC cone with the vertex at origin. This means the guiding curve cannot be of any arbitrary shape. $\endgroup$
    – Sasikuttan
    Mar 10, 2023 at 16:34
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    $\begingroup$ If the cone you're trying to identify is supposed to be a right circular cone, then you only need three points. However, if it can be any (elliptical) cone then you need five points. This can be justified from the degrees of freedom of their equations. $\endgroup$ Mar 10, 2023 at 17:15
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    $\begingroup$ @JeanMarie I tried some visualization in Geogebra, but my real issue was that I thought all cones are 2nd degree equations regardless of the shape of the guiding curve. It's clear now. $\endgroup$
    – Sasikuttan
    Mar 11, 2023 at 5:38

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