Calculating fundamental matrix Question about Projectin Matrices

I have a machine vision problem that requires me to calculate the Fundamental matrix from two images from unknown camera locations.

I'm assuming both cameras have ideal pinhole lenses for now.

The best solution I could find Is below:

My problem is with the transformation highlighted in red.

If the projection matrix P of the first camera maps the ray going from the center of the first camera through the point x in the image plane. And the projection matrix P' of the second camera maps the ray going from the center of the second camera through the point x' in the image plane.

Then how does P'P⁺x transform x to x'? Or it doesn't? Shouldn't it be P'TP⁺x where T is the transformation from Camera 1 to Camera 2?

Or does the Projection matrix include the world to camera transformation as well? I thought the projection matrix is independent of the camera location, it's an intrinsic property of the camera, correct?

Following this explanation is very confusing becasue the author doesn't specify what coordinate system he is using for each point/line. Does he change between Camera 1 and Camera 2's center as origin and rotation when talking about x and x'? Or does he maintain a fixed world coordinate system (which may be centered at camera 1, 2, or neither) for all points? Also, what is the [e']x notation? I cannot find this definition online anywhere.

My understanding when solving for fundamental matrix is to pick a few points (on a plane), find the homography between them, and this answer should be the rotational transformation between the cameras? Is this correct? Would I be able to find the translation using this homography as well? Or is this entirely incorrect?

Thanks for all your help, -D

Multiple View Geometry in Computer Vision Second Edition Richard Hartley and Andrew Zisserman, Cambridge University Press, March 2004. • You would be well advised to have a look at the entire book so that you know what the authors mean by “camera projection matrix” instead of (apparently) just diving into the middle of it. The matrices $P$ and $P'$ are the entire transformations from the 3-D object space to the 2-D space of the image, i.e., they are the compositions of the world and intrinsic matrices for each camera. Part of the point of these computations is that you don’t need to know the camera positions explicitly in order to relate the images. – amd Aug 16 '17 at 21:54
• Thanks, that makes more sense now. – Mich Aug 16 '17 at 22:20