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Given an ellipse with the following parameters:

  • $a$ = semimajor axis
  • $b$ = semiminor axis
  • $\theta$ = tilt angle from horizontal
  • $(\Delta x, \Delta y)$ = position of the center

How do I find the general formula of that ellipse, namely in the form

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

I’ve looked everywhere, and I can’t find anything even close to that.

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Stupid me! I actually found my answer by looking carefully into https://en.wikipedia.org/wiki/Ellipse#General_ellipse on Wikipedia… Thing is, I had found this answer before, but when computing it, I was coming to different results than the ellipse formula I was first given (to compare with). However, there may be many different formulas for the same ellipse (as they may simplify by dividing into factors, etc.), so I had overlooked that fact!

Thanks to user400188 and to Varun Vejalla…

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Here is pretty simple way to do it.

$\frac {((x-\Delta x)\cos\theta + (y-\Delta y)\sin \theta)^2}{a^2} + \frac {(-(x-\Delta x)\sin\theta + (y-\Delta y)\cos\theta)^2}{b^2} = 1$

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  • $\begingroup$ Yeah, but it’s not in the form that I was looking for: Ax²+Bxy+Cy²+Dx+Ey+F=0 $\endgroup$
    – Pierre Paquette
    Aug 13, 2020 at 2:16
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    $\begingroup$ You can always multiply it out... nice thing is that it is pretty clear how to build up the rotation, the translation and the dilation affects. $\endgroup$
    – Doug M
    Aug 13, 2020 at 2:17

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