How to get the properties of an ellipse with six points given.

Question

I am looking for a way to calculate the lengths of both semi-axes and the rotation angle of the ellipse in the image as shown in this picture. Six points are given, with two pairs of points being equidistant from the center and exactly in opposition. The other two points have the same x-position and their distance A1-A2 is also known. The diameters of the ellipse that end at the given points are conjugate as you can see in this question. Thank you for your help.

Answer 1

Thank you. I found a much easier way to get the properties of the ellipse using the Rytz's construction method.

Answer 2

Your task is a special case of the more general calculate ellipse from points, so my answer there applies here as well.

Since you have additional information, and even one more point than you'd need, you may make use of that information somehow. But I'd say the first step, the one for determining the rotation angle (using eigenvectors or a quadratic equation resulting from the application of rotation matrices), is always a good start. Once you have that, you can make use of the fact that you essentially already know the center, and don't have to read that off the equation of the ellipse.