How do you find the largest ellipse that can fit inside a quadrilateral tangent to the 4 sides of the quadrilatera?

Question

I am mainly curious if the ellipses artists practicing linear perspective put inside quadrilaterals are the largest possible ellipses they could be making given that same quadrilateral.

In 2-point perspective we find the tangent points by finding the perspective center by making an x inside the quadrilateral and then making a line from the vanishing point (A`) or the point where the two non-parallel lines of the quadrilateral would meet if extended. This finds two of the tangents. The other two are found by creating a line parallel with the horizon line through the perspective center point (F, O, U in the example).

Here are some examples of two point quadrilaterals (trapezoids/isosceles trapezoids).

in 3-point perspective we find the first two tangent points as we did in the first step of the two point quadrilaterals. This step must be repeated again for the other side as no two sides of the quadrilateral are parallel. This means there are two points where the lines created by extending the sides of the quadrilateral would meet (W1 & V1).

Do these tangent points create the largest possible ellipses inside the given quadrilaterals or are there other tangents that could create larger ellipses?

Answer 1

Just a quick graphical answer: exploring with GeoGebra one finds that, in general, "artist's ellipse" (obtained from a perspective transformation of a circle inscribed into a square, red in figure below) can have smaller area than another ellipse inscribed into the same quadrilateral (e.g. the blue one in the figure).