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I'm working of a set of datapoints known to be an elliptic paraboloid on which I best fit the general quadric $$ax²+bxy+cy²+dx+ey+f=0$$ Then I work with what I call radii projected on x an y defined as: $$R_x=-\frac{1}{2a}, R_y=-\frac{1}{2c}$$

Now I have a dataset for which the rotational term $bxy$ is far from neglectable and I would like to compute $Rx'$ and $Ry'$ along the ellipsoid natural axes. How can I do that? I suppose I should rewrite equation with something like this? $$u=x\cos \left(t\right)+y\sin \left(t\right)$$ $$v=x\sin \left(t\right)+y\cos \left(t\right)$$

AsukaMinato
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Julien M
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1 Answers1

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Looks like you've projected onto the $xy-$plane where the underlying figure is an ellipse.

Write your equation defining the ellipse as

$$\begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} a & \frac{b}{2} \\ \frac{b}{2} & c \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} d & e \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} +f=0 $$

Find the matrix $P$ that diagonalizes $A=\begin{pmatrix} a & \frac{b}{2} \\ \frac{b}{2} & c \end{pmatrix}$ so that

$$P^T A P = D.$$

Let $$\begin{pmatrix} x^\prime \\ y^\prime \end{pmatrix} = P \begin{pmatrix} x \\ y \end{pmatrix}$$

This gives new coordinates $(x^\prime, \, y^\prime)$ where the ellipse is in standard form.

mjw
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