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I have an ellipsoid as given here: $(p-C)^T Q(p-C)=1$

The terms $p, C$ and $Q$ are defined in the linked page.

My goal is to find equations for the $x,y,z$ co-ordinates of the resultant ellipsoid in a parametric form. For example, in this problem for an ellipse, I have the co-ordinates given by $\begin{aligned} {\left[\begin{array}{l}x \\ y \\ z\end{array}\right] } & =O^{\prime}+R\left[\begin{array}{l}x^{\prime} \\ y^{\prime} \\ z^{\prime}\end{array}\right] \\ & =O^{\prime}+\left[u_1, u_2, u_3\right]\left[\begin{array}{c}a \cos t \\ b \sin t \\ 0\end{array}\right] \\ & =O^{\prime}+a \cos t u_1+b \sin t u_2\end{aligned}$

I would like to parameterize my ellipsoid in quadratic form $(p-C)^T Q(p-C)=1$ to a parametric form similar to above. How can I achieve that?

To give more context: My aim is to extend the ellipse in this problem to an ellipsoid of revolution by revolving around the major axis.

wanderer
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If $Q$ is diagonal: $$ Q=\pmatrix{ 1/a^2 & 0 & 0 \\ 0 & 1/b^2 & 0 \\ 0 & 0 & 1/c^2 \\ } $$ then you can use the standard parametrisation: $$ x-x_C=a\cos t\sin s \\ y-y_C=b\sin t\sin s \\ z-z_C=c\cos s\phantom{\sin s} \\ $$ Otherwise you can diagonalize $Q$ with a rotation matrix $R$ (i.e. $R^TQR$ is diagonal) and then rotate back the above parametric equations: $$ \pmatrix{x-x_C \\ y-y_C \\ z-z_C}= R\pmatrix{a\cos t\sin s \\ b\sin t\sin s \\ c\cos s} $$

Intelligenti pauca
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