what is the difference between a projective mapping(transformation) and perspective mapping(transformation)

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

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

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

• Don’t try to look for consistency of terminology in different subdomains. For instance, “linear” means rather different things in different contexts.
– amd
Mar 17, 2020 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.

• 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.”
– amd
Mar 17, 2020 at 20:03

Slight correction necessary here I think where you said "a projective transformation is not necessarily a perspective transformation."

A projective transformation is by definition a perspective (transformation). That is, a projectivity is by definition a perspectivity. But geometric figures can be "projective" without being "perspective".

The ambiguities lie in the words: projective, perspective, projectivity and perspectivity.