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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.

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AplusKminus
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    As a general rule, you need $5$ points to define an ellipse. Indeed, the equation features $5$ independant parameters. – Martigan May 28 '15 at 13:44
  • Your canonical form is an ellipse, non-shifted, non-rotated and having radius 0, thus it is the point $(0,0)$. – mvw May 28 '15 at 14:09
  • What set of parameters do you need? The $A, \ldots F$ of $A x^2 + B xy + C y^2 + Dx + E y + F = 0$? – mvw May 28 '15 at 14:19
  • Or $a,b,\theta,C$ with $C$ being the center point and $\theta$ the orientation of the major axis. Any set that defines the ellipse would do. – AplusKminus May 28 '15 at 14:21

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