How to quantify the difference between $2$ ellipses?

Question

In my research I try to identify parameters defining an equation of ellipsoid. I have 3 unknowns and my ellipse is always centered at the origin but the rotation varies (see attached figure).

I would like to have some method for quantifying how good my identification was knowing the reference parameters.

Initially I thought of Coefficient of Determination, but I'm not sure if it's applicable to that problem. The values I obtain are very close to 1 (the ellipse shown in the figure has R2 = 0.990) so I'm not very confident if this tool is reliable to distinguish between different sets of parameters.

Another idea was to compare areas of each ellipse, but here the problem lies in rotation - the area is conserved through rotation, but clearly two rotated ellipses are not the same. The area between the ellipses is another idea, but here I struggle with integration limits.

Any suggestions how reliably could I quantify accuracy of my parameters?

EDIT: My model is based on 3 parameters: $\sigma_{11}^{y}, \sigma_{22}^{y}, \sigma_{33}^{y}$ And the curve (in $\sigma_{11} - \sigma_{22}$ space) I construct is as follows:

$\left(\frac{\sigma_{11}^{y}}{\sigma_{11}^{y}}\right)^{2} \sigma^{2}_{11} -\left[\left(\frac{\sigma_{11}^{y}}{\sigma_{11}^{y}}\right)^{2} +\left(\frac{\sigma_{11}^{y}}{\sigma_{22}^{y}}\right)^{2}-\left(\frac{\sigma_{11}^{y}}{\sigma_{33}^{y}}\right)^{2}\right] \sigma_{11} \sigma_{22} + \left(\frac{\sigma_{11}^{y}}{\sigma_{22}^{y}}\right)^{2} \sigma^{2}_{22} - \left(\sigma_{11}^{y}\right)^{2} = 0$

Answer 1

As I mentioned in the comment, one way to measure the difference is to integrate (in polar coordinates, e.g., with the origin coinciding with the center of both ellipses) the absolute value of the difference of how far each point lies away from the center.

In other words, if we consider $f(\theta)$ to be the distance of one ellipse from the origin in the angle of $\theta$, and $g(\theta)$ - of the other, we need $$ D_{f,g} = \int_0^{2\pi} \left\|f(\theta) - g(\theta)\right\| \mathrm{d}\theta, $$ where the norm could be $2$-norm or $1$-norm or any other convenient norm.

Answer 2

If rotation was the only parameter that varied, then the only thing that would changes in your given equation is the coefficient of the $\sigma_{11}\sigma_{22}$-term, hence you can compare how close they are by comparing that coefficient. However, you also have that the coefficient of $\sigma_{22}^2$ can change, as well as the constant term (which is $-(\sigma_{11}^y)^2$). Note that the coefficient of the $\sigma_{11}^2$-term is always $1$.

If we call $x=\sigma_{11}$, and $y=\sigma_{22}$, then your equation has the general form:

$x^2+Bxy+Cy^2+F=0$

Where

$B=-\left[\left(\frac{\sigma_{11}^{y}}{\sigma_{11}^{y}}\right)^{2} +\left(\frac{\sigma_{11}^{y}}{\sigma_{22}^{y}}\right)^{2}-\left(\frac{\sigma_{11}^{y}}{\sigma_{33}^{y}}\right)^{2}\right]$

$C=\left(\frac{\sigma_{11}^{y}}{\sigma_{22}^{y}}\right)^{2}$

and

$F=- \left(\sigma_{11}^{y}\right)^{2}$

$B$ controls rotation, $C$ the length ratio between major and minor axis ($C>0$ because of the square appearing on the RHS above, and nonzero because $\sigma_{11}$ must be nonzero else the result isn't an ellipse), finally $F$ controls the size of the ellipse (to see why, consider $x=0$, then the above equation becomes $Cy^2-F=0$)

The only question now is: How exactly do you want to 'put a number' to the error. Or in other words, how do you want to measure the error