In my particular case I am given a projection of a circle onto the $xy$-plane and the radius $r$ of said circle. This results in an ellipse with semi-major axis $a$ equal to $r$.
Like in this other question I want to find the orientation of the circle -- meaning the normal vector and its origin. But contrary to the linked question, I cannot determine the bounds of the ellipse, as my image contains too much noise. Therefore I first need to fit an ellipse to the image. I plan to do this via the RANSAC method. So you can consider $n$ points on the ellipse as given (with $n$ being just as large as it needs to be).
As I don't know the orientation of my ellipse, the canonical form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ doesn't help me, so I assume I'll need the analytical form:
$$px^2+qxy+ry^2+sx+ty+u=0$$
The semi-major axis $a$ does not appear in this formula and is also not directly involved in any of the constants -- at least not without also knowing the center point of the ellipse (see this for details). How many points do I need to fix the ellipse? How do I calculate any set of parameters that defines the ellipse?
What I have:
- $a$: The length of the semi-major axis
- $P$: A set of at least $n$ points $p_i(x_i|y_i)$ on the ellipse
- A method to determine whether the calculated ellipse fits the image
What I do not have:
- The center point of the ellipse
- The orientation of the ellipse
- The semi-minor axis
What I need:
- $n$: The number of points necessary to fix the ellipse. I suspect it to be 3, because $a$ is given, but I am not sure.
- $S$: A set of parameters defining the ellipse, dependent on $a$ and $P$.
Assume a general non-degenerate ellipse for the answer.