What is the minimal number of points $N$ to uniquely define the semi-major axis $a$, the semi-minor axis $b$ and the rotation angle $\omega$ of an ellipse whose the center is known/fixed (this is related to ellipse fitting procedures).
In other words, if we consider that the center is known/fixed, on this image what is the minimal number of blue points I need to compute $a$, $b$ and $\omega$ in a unique way ?
EDIT: and what is the number of points needed when the center of the ellipse is not known ?
Best Answer
The equation of an ellipse is given by $$ ax^2+by^2+cxy+dx+ey+f = 0 $$ where $(a,b,c,d,e,f)\in\mathbb{P}^5(\mathbb{R})$, hence you need $5$ points to recover the coefficients, if you do not know where the center is. Otherwise, $3$ points (such that no two of them are symmetric with respect to the center) are enough, since their mirror images with respect to the center of the ellipse still lie on the ellipse, so having $3$ points and the center is like having $6$ points on the ellipse. Obviously, having two points and the center is not enough, since there are many concentric ellipses that pass through the same two points.