Projective transformations of the plane

A projective transformation of the plane is a special kind of mapping that takes points, lines, and other geometric objects in the plane and moves them to new positions, while preserving straightness (lines map to lines) and the property that three points are collinear if and only if their images are collinear. Projective transformations are more general than similarities or affine transformations, because they can 'move' points at infinity and change parallel lines into intersecting lines.

Important properties

• Projective transformations map lines to lines.

• Collinearity of points is preserved.

• They can send parallel lines to intersecting lines (and vice versa).

• They are often represented using matrices acting on homogeneous coordinates.

• They preserve the cross ratio of four collinear points.