Projective transformation

A projective transformation is induced by an invertible linear map on homogeneous coordinates. It maps lines to lines but need not preserve parallelism, lengths, or angles.

Homogeneous coordinates: Coordinates that represent a point by any nonzero scalar multiple of the same coordinate tuple. They let an invertible matrix act on points, including points at infinity.
Perspective projection: A mapping that projects three-dimensional points onto an image plane from a common viewpoint. Its planar effects are described by projective transformations.
Projective geometry: The geometry of incidence relations among points, lines, and planes, independent of metric measurements. Projective transformations are its coordinate representations of incidence-preserving changes.
Euclidean transformation: A rigid motion or reflection that preserves Euclidean distances and angles. Projective transformations generally preserve neither distances nor angles.
Projective matrix: A nonsingular square matrix representing a projective transformation, defined up to nonzero scalar multiplication. Its action on homogeneous point coordinates is the transformation itself.
Homography: A projective transformation between two projective planes, commonly represented by a 3-by-3 matrix. It models the image-to-image mapping between views of a planar surface.
Projective plane: A geometric plane augmented with points at infinity so every pair of distinct lines intersects. It is the natural domain for planar projective transformations.
Similarity transformation: A map combining a Euclidean transformation with uniform scaling. Unlike a general projective map, it preserves angles and shapes up to scale.
Composition of projective transformations: The operation of applying one projective transformation after another; their representing matrices multiply. Matrix multiplication makes successive perspective changes another projective transformation.
Camera calibration: The process of estimating a camera’s internal parameters and its pose from image observations. Projective mappings connect calibration patterns to their observed image points.

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