# How do I convert an expression in terms of the general equation of a conic section to one in the equation of an ellipse?

In a major assignment I am to determine the semi-major axis of an elliptic orbit for the star S2 around Sagittarius A*. I found some data that I have used to fit the points to an ellipse - however the equation I get is in terms of

$$ax^2 + bxy + cy^2 + dx + ey + f = 0$$

rather than

$$\frac{x^2}{a^2} + \frac{y^2}{ b^2}= 1$$.

Can any of you smart people teach me how to 'translate' it, or if it's even necessary in order to determine the semi-major axis of the ellipse?

• Where is your data relative too, Sagittarius A*?
– Paul
Dec 12, 2018 at 13:00
• Yes. It's measured in arc seconds of declination and right ascension relative to SgrA*. Dec 12, 2018 at 13:05
• Well, the first problem is that in the second equation, it's assumed that the ellipse is located at the origin of the coordinate system. Therefore, you have to move it to the right location first ... Dec 12, 2018 at 13:05
• I'm not very adept when it comes to ellipses, but I think it is located at the origin of the coordinate system as (0,0) is located at one of the foci. Dec 12, 2018 at 13:08
• Another problem is that declination and right ascension are spherical coordinates and are measured in different units, so there's a question of whether "one arc-second" measures the same apparent distance in both directions. Also, it seems we are not looking "straight down" on S2's orbit but rather view it from an oblique angle, which means the major axis of its path on the celestial sphere may be very different from the major axis of the actual orbit. The distance between two points on the orbit depends on the change in distance from us as well as the apparent angle. Dec 12, 2018 at 13:33

Think of the ellipse as a quadratic form: $$[x,y,1]\begin{bmatrix}a&b/2&d/2\\b/2&c&e/2\\d/2&e/2&-f \end{bmatrix}\begin{bmatrix}x\\y\\1 \end{bmatrix}=0$$. Lets call the matrix in the middle $$Q_0$$ and then $$[x,y,1]Q_0 \begin{bmatrix}x\\y\\1 \end{bmatrix}=0$$

Since you are allowed translate and rotate the coordinate system, you can push a 2d "rigid motion" matrix $$E$$ such that $$[x,y,1] E^T Q_1 E\begin{bmatrix}x\\y\\1 \end{bmatrix}=0$$ and you can look for such and $$E$$ such that $$Q_1=\begin{bmatrix} a'&0&0\\0&b'&0\\0&0&-1 \end{bmatrix}$$

The general form of $$E$$ is $$\begin{bmatrix}\cos(\theta)&\sin(\theta)&p_x \\-\sin(\theta)&\cos(\theta)&p_y\\ 0&0&1 \end{bmatrix}$$

• In such issues (attraction by a heavy mass), it is preferable to use polar representation $p=p_0/(1+e \cos(\theta))$ where the center is the center of mass (math.stackexchange.com/q/2389034) Dec 12, 2018 at 20:48
• Why is it preferable? Dec 13, 2018 at 5:36
• if you look for example to the way the 3 Kepler's laws are derived from the (Newton's) inverse square law of gravitation, you will see that it's this form which is used. See for example formula $r(\theta)=...$ on page 6 of this document math.utk.edu/~freire/teaching/fall2006/m142f06NewtonKepler.pdf Dec 13, 2018 at 7:25
• In fact, I wanted to make this comment to the OP, not to you... Dec 13, 2018 at 7:26
• I think that for regression fitting purposes, it is in fact more convenient to keep the $x,y$ Cartesian form, but to to each his own. Dec 13, 2018 at 10:11