How to obtain *A*, *B* and *C* from the equation of and oblique ellipse

I have the equation of an ellipse centred at the origin and inclined to the coordinate axes: $$\frac{(x\cos\theta + y\sin\theta)^2}{a^2} + \frac{(y\cos\theta - x\sin\theta)^2}{b^2} = 1$$ In order to find the rotation angle I know that: $$\tan(2\theta)=\frac B{A-C}\$$ Now my problem is to obtain the forms of A, B and C for the latter equation, from the former. Any help will be very welcome.

• What's the definition of A, B and C? – NickD May 4 '17 at 18:04
• I think it is $A x^2 + B y^2 + C x y + F=0$ and you want to get to $A' x'^2 + B' y'^2 = 1$ Is this correct? – ja72 May 4 '17 at 19:33
• it is correct. I wrongly wrote x' and y' when I meant x and y, but in any case it is the same problem. I want to find A, B and C corresponding to the coefficients of the general equation. – P Macmutton May 4 '17 at 20:02

All you need is to explicitly write your first equation by expanding the squares and grouping terms for $x'^2$, $y'^2$, and $x'y'$. Their coefficients are $A$, $C$, and $B$. You will be surprised to see $$\tan(2\theta)=\frac{2\sin\theta\cos\theta}{\cos^2\theta-\sin^2\theta}$$ :)

• Thanks. I did so but couldn't get to the same form. According to your answer A = cos²(theta); C = sin²(theta); and B = 2sin(theta)cos(theta). What happens to coefficients a and b? – P Macmutton May 4 '17 at 20:05
• Sorry, just now I realized what you meant in the fraction is sin2(theta) / cos2(theta)... Thanks again, but I still do not get how to obtain the forms of A, B and C... – P Macmutton May 4 '17 at 20:17
• $B=2 \sin \theta \cos \theta\left(\frac{1}{a^2}-\frac{1}{b^2}\right)$. $A$ and $C$ contain the same expression in $1/a^2-1/b^2$, which will cancel if $a\ne b$. But if $a=b$ then your ellipse is a circle, so there is no tilt – Andrei May 4 '17 at 20:18
• Explicitly find terms for $x^2$, $y^2$, and $xy$. In the first equation, if you expand the squares, you get the coefficient for $x^2$ to be $A=\frac{\cos^2\theta}{a^2}+\frac{\sin^2\theta}{b^2}$. My previous comment wrongly stated that $A$ and $C$ have the same expression in $a$ and $b$. In fact only the difference $A_C$ has the same expression – Andrei May 4 '17 at 20:24
• Thanks again Andrei, now I arrived to the same, and C=\frac{\sin^2\theta}{a^2}+\frac{\cos^2\theta}{b^2} – P Macmutton May 4 '17 at 20:55

In the world coordinates you have $$A x^2 + B y^2 + C x y - 1 = 0$$ and you want to find the parameters of the ellipse in a coordinate system that is aligned with the major/minor axis. We rotate the local coordinated by $\theta$ with:

$$x = x' \cos\theta -y' \sin\theta \\ y = x' \sin\theta + y' \cos\theta$$

If you plug it in the ellipse equations you will find from the terms multiplying $x y$ that $$\left. 2 C \cos^2 \theta + 2 (B-A) \sin \theta \cos\theta - C =0 \right\}\\ \left. C \cos(2 \theta)+(B-A) \sin(2 \theta) = 0 \right\}\\ \tan (2 \theta) = \frac{C}{A-B}$$

Use this angle to get the aligned ellipse

$$\left( \tfrac{\sqrt{ (A-B)^2+C^2}+A+B}{2} \right) x'^2 + \left( \tfrac{-\sqrt{(A-B)^2+C^2}+A+B}{2} \right) y'^2 = 1$$