Can every arbitrary set of four points in 2D space map to at least one right-angled rectangle perspective-projected from 3D space onto the plane?

Question

I have an intuition that given a rectangle of arbitrary width and height rotated arbitrarily in 3D space and perspective-projected onto the 2D plane, that not all arbitrary sets of resulting 2D points are possible.

That is to say that I intuit that there is a constrained relationship between the points due to the original rectangle having the constraints of right angles and parallel sides.

Is my intuition right or wrong?

Answer 1

I've done a lot of searching and reading on the web since posting the question but I don't understand all the concepts in all the articles so I may have overlooked something...

It turns out my intuition was only "slightly right", but mostly wrong.

There are only two constraints if the 4 points are not considered to have an intrinsic order:

1. The four points must form a convex quadrilateral.
2. No three points can lie on the same line. (The degenerate case where the points actually form a triangle. Not illustrated.)

I didn't think about this when posting the question, but if the points are considered to have an intrinsic order then there is a constraint that no two edges may cross one another. But this is just one kind of non-convex quadrilateral anyway, so is already covered.

My intuition was that there would be a lot more to constrain acceptable sets of four points than this, so in that I was wrong.

I now believe that for any given set of four points forming a convex quadrilateral there are exactly two squares in 3D space than can map to them via some perspective projection onto the 2D plane, and an infinite number of rectangles. (I did not fully understand every concept in the articles I read to reach this conclusion though, so I may still be in error.)