Projective space

Projective space is the set of lines through the origin in a vector space, with each line represented by one point. It extends affine space by including directions as points at infinity.

Homogeneous coordinates: Coordinates defined up to multiplication by a nonzero scalar, used to represent points in projective space. They encode each line through the origin as a coordinate tuple.
Projective transformation: A transformation induced by an invertible linear map on the underlying vector space, considered up to scalar. Such maps preserve the incidence structure of projective space.
Perspective projection: A mapping that projects points from a viewpoint onto an image plane, producing perspective effects. It is naturally expressed as a map between projective spaces.
Incidence geometry: The study of which points, lines, and other subspaces are incident with one another. Projective space supplies a central setting for incidence questions.
Equivalence relation: A relation that partitions a set into classes of elements treated as equivalent. Projective points are equivalence classes of nonzero vectors under scalar multiplication.
Projective coordinates: Coordinates for projective points in which nonzero scalar multiples represent the same point. They provide a calculational language for points and transformations.
Pinhole camera model: An ideal camera model in which light passes through a single point to form an image on a plane. Its image formation equations use homogeneous projective coordinates.
Grassmannian: A space whose points represent linear subspaces of a fixed dimension in a vector space. Projective space is the Grassmannian of one-dimensional subspaces.
Affine space: A geometric space modeled on a vector space but lacking a distinguished origin. Projective space adds ideal points to affine space while preserving its finite points.
Projective line: The one-dimensional projective space formed by the lines through the origin in a two-dimensional vector space. It makes the added point at infinity explicit in the simplest projective space.

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