Projective geometry

Projective geometry studies properties preserved by projective transformations, especially incidence and cross-ratio. It does not require distance or angle measurements.

Projective plane: A geometric plane in which every pair of distinct lines meets at exactly one point. It provides the setting where parallel lines meet at points at infinity.
Projective frame: A minimally sufficient set of points that fixes coordinates and determines a projective transformation. Frames show how projective transformations are fixed by finite geometric data.
Girard Desargues: A seventeenth-century French mathematician whose work helped establish projective methods in geometry. His theorem became a foundational result about projective incidence.
Perspective (graphical): A method of depicting three-dimensional scenes on a two-dimensional surface from a viewpoint. Perspective drawing is a direct visual instance of projective mapping.
Euclidean geometry: Geometry based on distance, angle, and the parallel postulate in flat space. Projective geometry retains incidence but not Euclidean lengths or angles.
Homogeneous coordinates: Coordinates defined up to nonzero scale, allowing points at infinity to be represented alongside ordinary points. They encode projective points and transformations with linear algebra.
Collineation: A bijection of a projective space that maps lines to lines. Collineations express projective transformations through their action on incidence.
Blaise Pascal: A French mathematician whose theorem describes a hexagon inscribed in a conic. His conic theorem became a landmark of early projective geometry.
Projective camera model: A mathematical model mapping three-dimensional points to image coordinates through perspective projection. It applies projective transformations to image formation.
Affine geometry: Geometry that preserves collinearity, parallelism, and ratios along a line, but not general lengths or angles. Affine geometry preserves parallelism, which projective transformations need not preserve.

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