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

Question

(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 ?

Answer 1

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.

Answer 2

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.