Perspective Ellipse Math for SoK'22

[Srirupa Datta]

I have a quadrilateral, whose 4 corners are known, and an inscribed ellipse
whose touching points with the quadrilateral and also known. I need to find
out the extreme points of the major axis of that ellipse.

*Method 1*
I decided to first find out the equation of the inscribed ellipse. This
<https://drive.google.com/file/d/1mMT9to50lJN1eo2SPQh3jJB_puvb-SBj/view?usp=sharing>
is what I did so far. However this approach has a challenge. I need to
solve a system of linear equations for six variables and this is *probably*
complicated. So I'm not sure if this is at all the right approach.

Also, I got stuck since I expected the solution to be a unique one.

*Method 2*
I have the transformation matrix that maps a unit square to my
quadrilateral. Is there any way I can use transformation mathematics to
solve this problem easily?

Thanks,
Srirupa
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