# Calculate Ellipse From Points?

How can I calculate an ellipse from a group of points ?

I'm not mathematician so I don't really know the best parameter style for ellipses. This ellipse is aligned to X and Y axes and I usually go with center point and 2 radii in x and y directions.

# Finding parameters

An ellipse (and in fact any conic section) is described by an equation of the form

$$ax^2+by^2+cxy+dx+ey+f=0$$

Any multiple of this equation describes the same ellipse. The parameters $a$ through $f$ can be found up to that multiple by knowing $5$ points on the ellipse:

$$\begin{pmatrix} x_1^2 & y_1^2 & x_1y_1 & x_1 & y_1 & 1 \\ x_2^2 & y_2^2 & x_2y_2 & x_2 & y_2 & 1 \\ x_3^2 & y_3^2 & x_3y_3 & x_3 & y_3 & 1 \\ x_4^2 & y_4^2 & x_4y_4 & x_4 & y_4 & 1 \\ x_5^2 & y_5^2 & x_5y_5 & x_5 & y_5 & 1 \\ \end{pmatrix} \cdot \begin{pmatrix} a \\ b \\ c \\ d \\ e \\ f \end{pmatrix} =\begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}$$

If you have more than five data points, you may choose any five of them. If your data points are not perfectly accurate, then look into Ellipse fitting methods.

# Changing the representation

To turn the parameters you found into center, radii and orientation you can follow the steps which I outlined in this answer.

• I edited the matrix which has slips in the first and second columns (supposed to be $x_1^2$, not $x_1$, etc.). Now they are fixed. – C. Y. Cheng Apr 24 '14 at 20:15