2

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.

Varun Vejalla
  • 9,780
Pierre Paquette
  • 235
  • 2
  • 14
  • This answer comes very close to what you are asking and may be of help: https://math.stackexchange.com/questions/1217796/compute-center-axes-and-rotation-from-equation-of-ellipse/1217797#1217797 (1 of 2) – user400188 Aug 13 '20 at 01:36
  • However, I do not think that this question is a duplicate, as the answer I found (and other answers of a similar nature https://math.stackexchange.com/questions/426150/what-is-the-general-equation-of-the-ellipse-that-is-not-in-the-origin-and-rotate) do not give the generalised equation for a rotated ellipse in terms of the variables you ask for. (2 of 2) – user400188 Aug 13 '20 at 01:39

2 Answers2

2

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…

Pierre Paquette
  • 235
  • 2
  • 14
2

Here is pretty simple way to do it.

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

Doug M
  • 58,694
  • Yeah, but it’s not in the form that I was looking for: Ax²+Bxy+Cy²+Dx+Ey+F=0 – Pierre Paquette Aug 13 '20 at 02:16
  • 1
    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. – Doug M Aug 13 '20 at 02:17