How to find the axes of a rotated ellipse

Question

I have a set of 17 points which I know are on an ellipse. I have the x,y co-ordinates of each point; the y-axis is vertical and the x-axis is horizontal. I want to prove these points are on an ellipse, but the ellipse is rotated clockwise by approximately 14 degrees (determined visually - I want to calculate the exact value of the rotation). I need to find the exact position of the major and minor axes (x' and y') and I do not know the values of the semi-major axis (a) or the semi-minor axis (b). Is this possible?

I have tried to find the x,y co-ordinates of the points furthest from (and nearest to) the centre of the ellipse, but I've only managed this through very many tedious iterations and it isn't exact -- is there a better way? Thank you.

Answer 1

An ellipse on the $xy$ plane is uniquely determined by five independent parameters: $B, C, D, E, F$ such that $ x^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $ with the restriction that $ B^2 - 4C < 0$. So, you need to solve for $5$ unknown variables in a linear equation, which requires at least $5$ pairs $(x, y)$ such that the corresponding $5$ vectors $(xy, y^2, x, y, 1)$ are linearly independent.

From your observations, choose any $5$ $(x, y)$-pairs such that the $(xy, y^2, x, y, 1)$ vectors are linearly independent. Then input the values of $x^2, xy, y^2, x, y$ in $ x^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $ to form $5$ linear equations in $B, C, D, E, F$, and solve this system of linear equations to obtain the values of $B, C, D, E, F$.

Note that in the link to the Wikipedia page, there are $6$ parameters described. However, these $6$ parameters are not independent and the ellipse can only have $5$ independent parameters. So, the parameter $A$ is set to $1$ above. This amounts to dividing each of the other parameters by $A$. Since $A$ is the coefficient of the term $x^2$ in the equation of the ellipse, it cannot be zero.

The rotation $\theta$, the possible linear shift of the origin to $(x_0, y_0)$, the major and lengths of the major axis $2a$ and the minor axis $2b$ can be computed from these $5$ parameters $B, C, D, E, F$ (with setting $A=1$), as defined in this page.