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?