Testwiki:Reference desk/Archives/Mathematics/2011 August 16

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August 16

Extracting parameters from a matrix form ellipse

I found an algorithm for fitting the smallest possible ellipse through a set of points, but the solution is provided as the set of points $𝐩$ for which $(𝐩-𝐜{)}^{\mathrm{T}}𝐀(𝐩-𝐜)=1$ where $𝐀$ is a symmetric, positive definite matrix.

$𝐜$ is already useful as the center of the ellipse, but in order to plot it I need the whole thing in parametric form. This means I need to decompose $𝐀$ into some parameters for the parametric form:

$X(t)=X_c+a\cos t\cos \phi -b\sin t\sin \phi$

$Y(t)=Y_c+a\cos t\sin \phi +b\sin t\cos \phi$

Is there a simple formula for extracing $a$, $b$ and $\phi$ from $𝐀$?

Eonzo (talk) 10:08, 16 August 2011 (UTC)

If you write $𝐩-𝐜=\left[\begin{array}{c}x\\ y\end{array}\right]$ and $𝐀=\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]$, then $(𝐩-𝐜{)}^{\mathrm{T}}𝐀(𝐩-𝐜)=a{x}^2+2bxy+c{y}^2=1$. To finish up, see Rotation of axes#Elimination of the xy term by the rotation formula. 130.76.64.117 (talk) 00:45, 17 August 2011 (UTC)

Thank you for pointing me there, that's exactly what I needed :) Eonzo (talk) 07:51, 17 August 2011 (UTC)