. Why can a homography (3x3 matrix on homogeneous coordinates) represent translation, while a plain 2x2 matrix cannot?
Representing a 2D point as (x, y, 1) lets a 3x3 matrix encode translation directly, unlike a 2x2 matrix acting on (x, y) alone.
. What specifically causes a homography to destroy parallelism, unlike an affine transform?
An affine transform's bottom row is always (0, 0, 1); a homography's nonzero (g, h) introduces the division effect that makes parallel lines converge.
. How does the point-at-infinity trick locate a vanishing point?
A point with zero third homogeneous coordinate represents a direction at infinity; transforming it by H lands exactly where the two parallel lines' images converge.
. In what two situations does a single homography exactly relate two photographs of a scene?
A homography exactly models plane-to-plane relationships (a flat scene) or pure camera rotation about a fixed center; a translating camera viewing a genuinely 3D scene has no single exact homography relating the views.