August 16

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

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

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

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

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

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

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

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