Computing the properties of the 3D-projection of an ellipse.

I have an ellipse that is rotated around the white axis (see image below) in 3-dimensional space by an angle α. The axis passes through the perimeter and one of the ellipse's focal points F. In the picture the angle of the original blue ellipse E is 0°. When rotated by 180° it would be mirrored to the other side of the axis. When rotated by 90° it would point away from the viewer (P1 behind the axis), rotated by 270° it would point towards the viewer (P1 in front of the axis). The red ellipse E' is being rotated by 60°. The positions of points P1' to P4' and of point M' can be easily computed using the cosine of α.

But to actually draw the ellipse these points are useless to me. What I need are the white points, that are the end points of the two main axes of the ellipse E' being projected onto the plane of ellipse E. But I did not find a way to compute the values of a', b' and β' in relation to the angle of rotation α (a, b and β are known). I would really appreciate if anyone could help. Thank you.

• Whith 4 points you can easily find the properties of the ellipse. Are you looking for the locus of the vertices of the ellipse as it is rotated, instead? – N74 May 3 '16 at 8:48
• I have the properties of the vertices of both ellipses as lengths and angles from a central point and also as positions in a cartesian coordinate system. So I could use these to feed them to a formula that gives me either the positions of the new vertices or the lengths of both semi-axes and the angle β', that is both equally good. Could you point me to a resource that explains how to do it? I found this but I don't understand it. Thanks. – atarax42 May 3 '16 at 14:43