I have a doubt on the classification of non-degenerate conics (parabola, ellipse, hyperbola) in projective geometry (my textbook is "Multiple View Geometry in Computer Vision", which, as the title implies, is not specifically targeted at projective geometry).
As far as I understand, we can classify conics from a projective and from an affine point of view:
From the projective point of view, all non-degenerate conics can be transformed to a circle, so there's no distinction between the three "classic" conic types.
From the affine point of view, the classification depends on the number of intersections (0: ellipse, 1: parabola, 2: hyperbola) between the conic and the line at infinity, which is fixed for affine transformations.
Summarizing, it seems that a generic projective transformation alters the number of intersections between a conic and the line at infinity, whereas an affinity does not change it.
My question is: why does the number of intersection between a conic and the line at infinity change for a projectivity? Shouldn't intersection be an invariant?
Thanks in advance!