Here are some attempts to illustrations:


Cricles.


From a perspective



Three ellipses (or halves thereof) for which points are available are coloured for illustration. There will be three distinct sets of points to fit. Red points for the red ellipse and so on. For example, no red point is to be used in fitting the green ellipse.

Generally, the fragment of each ellipse available, is a bit too small for reliable parameter estimation. But if they could all be put together and fitted at once, the estimates should become much better.

If it is of any help, here is some of my reasoning about the relationship between circles and ellispes. It is difficult for me to describe this, and I'm not 100% it is correct, but I make a try:

Introduce an imaginary circle at right angle to a circle observed. The two circles intersect in two points. The diameter between these two points forms the vertical axis of the ellipse for this circle. I introduce two triangels. The angle at the center of the imaginary circle is the same as the angle at the "viewing point" on the periphery of the same imaginary circle, no matter where on the imaginary circle the viewing point is located. And this angle is proportional to the length of the vertical axis of the ellipse seen from the viewing point.


A new line is drawn from the viewing point to the "bottom" point of the imaginary circle, right under its center. Now consider the point where this line intersects the diameter of the circle. This is where we find the center of the ellipse, so it is how the length of the horisontal axis of the ellipse is determined. The length of the horisontal axis is proportonal to the viewing angle in the corresponding triangle.

So, the ratio between the lengths of the two axes of an ellipse, is (I think) the same as the ratio between those two angles at the viewing point which I have tried to describe.

Each circle has its own "imaginary circle". But all these imaginary circles intersect in the same "viewing point". I think that this is the geometry one has to use in order to translate the relationship between the circles in 3D space, to a relationship between the parameters of the ellipses in the 2D plane where they are "projected".

I write "vertical" and "horisontal" here, but the axes are tilted at some angle.