# Ellipse general equation from dimensions, offset, and tilt angle

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.

• This answer comes very close to what you are asking and may be of help: math.stackexchange.com/questions/1217796/… (1 of 2) Aug 13, 2020 at 1:36
• However, I do not think that this question is a duplicate, as the answer I found (and other answers of a similar nature math.stackexchange.com/questions/426150/…) do not give the generalised equation for a rotated ellipse in terms of the variables you ask for. (2 of 2) Aug 13, 2020 at 1:39

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