Calculating Epipolar geometry, does 7 or 8 point algorithm require points to be complanar

Question

I'm doing some calculations based on a few documents I found online:

http://dhoiem.cs.illinois.edu/courses/vision_spring10/lectures/Lecture22%20-%20Epipolar%20Geometry.pdf

https://www.robots.ox.ac.uk/~vgg/hzbook/hzbook1/HZepipolar.pdf

My question is, according to the definition of a fundamental matrix, it's supposed to turn any point correspondences between two stereo images into epipolar lines.

Therefore, the Fundamental matrix should inherently be the same for any two matching points between two images. Why else would they call it Fundamental, right?

So I ran a test, where I rendered with two cameras two rectangles, 8 points, 4 of which which were not co-planar to the other 4, and did Gaussian elimination to find entries for the fundamental matrix.

My problem is that the Determinant of the matrix is not 0

However, I ran the test again, this time, I made the two planes co-planar, so all of the 8 points were now co-planar.

This resulted in a fundamental matrix with a determinant of 0

My question is, when finding the fundamental matrix using either the 8 or 7 point algorithms, do all points have to lie on the same plane??

All the examples I have seen online seem to use completely non-coplanar points. For example: Example

Thanks,

Answer 1

I contacted a couple specialists with this specific question, and I got the answer that the points do not have to lie in the same plane.

However, they do need to be normalized, and that is perhaps where my calculation error was