(1)a projective but not perspective mapping indicates that they are different

(2)Perspective Transform & Homography Matrix indicates these are the same thing

(3) projective transformation = homography = collineation

it is evident (1) conflicts with (2) and (3). how can pinpoint these ?

  • $\begingroup$ Don’t try to look for consistency of terminology in different subdomains. For instance, “linear” means rather different things in different contexts. $\endgroup$ – amd Mar 17 at 20:06

In projective geometry a projective transformation is a product of perspective transformations. A perspective transformation is a projective transformation, but a projective transformation is not necessarily a perspective transformation.

A projective transform is a homography is a collineation.

In general, the transformation between four corresponding pairs of points is a projective transform.

The blog post (2) gets it wrong. The OpenCV's getPerspectiveTransform function seems to be incorrectly named. It should be called getProjectiveTransform, I suppose, but presumably nobody in that community objects.

So it's actually (2) that conflicts with (1) and (3), and I'd venture that's because (1) and (3) are math while (2) is computer vision software, where terminology may differ. It could be that in computer vision the most common use of a projective transform is to remove or add perspective.

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  • $\begingroup$ It’s common in computer graphics to call a plane homography a “plane projective transform.” To muddy the waters further, the mapping from $\mathbb P^3$ to $\mathbb P^2$ that models a pinhole camera is also called a “projective transformation.” $\endgroup$ – amd Mar 17 at 20:03

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