In Scott Robertson's How to Draw book he proposes a method for finding proportional squares using ellipses:
(the "conditions" he mentions are described on the previous page and are simply that the left and right sides of the ellipse should meet the sides of the plane at their midpoints and the top and bottom points should be vertically aligned)
I understand that a circle always becomes an ellipse in perspective, but I don't understand the idea being proposed here that an ellipse fit to the side of a plane in perspective must be a perfect circle in plan view. What is to stop this ellipse being drawn here from also being a (differently proportioned) ellipse in plan view?
Through experimentation I have found this to be true though. Below are two perfect cubes in a 3D modelling program, and when overlaying the ellipse tool over both of them the only way to get the ellipse to fit in the way described by Scott ends up with the ellipse fitting perfectly on the face of one of the cubes.
However, I have also found one edge case where this isn't true. When the ellipse minor axis is parallel to the horizon line any sized ellipse fits within the rules of what should create a perfect square.
By Scott's rules this is a perfect square:
And yet so is this:
So since this rule does indeed hold true in some cases, but not in all, my questions are:
1) What is the reasoning behind this rule? How does drawing an ellipse in this way result in a perfect circle?
2) When can this rule be applied, and when does it break down?