Ellipse general equation from dimensions, offset, and tilt angle

Question

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.

Answer 1

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…

Answer 2

Here is pretty simple way to do it.

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