Wikipedia: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 $\mathbf {p}$ for which $(\mathbf {p} -\mathbf {c} )^{\mathrm {T} }\mathbf {A} (\mathbf {p} -\mathbf {c} )=1$ where $\mathbf {A}$ is a symmetric, positive definite matrix.

$\mathbf {c}$ 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 $\mathbf {A}$ into some parameters for the parametric form:

$X(t)=X_{c}+a\,\cos t\,\cos \varphi -b\,\sin t\,\sin \varphi$

$Y(t)=Y_{c}+a\,\cos t\,\sin \varphi +b\,\sin t\,\cos \varphi$

Is there a simple formula for extracing $a$, $b$ and $\varphi$ from $\mathbf {A}$?

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

If you write $\mathbf {p} -\mathbf {c} ={\begin{bmatrix}x\\y\end{bmatrix}}$ and $\mathbf {A} ={\begin{bmatrix}a&b\\b&c\end{bmatrix}}$, then $(\mathbf {p} -\mathbf {c} )^{\mathrm {T} }\mathbf {A} (\mathbf {p} -\mathbf {c} )=ax^{2}+2bxy+cy^{2}=1$. 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)