Homogeneous coordinates

Homogeneous coordinates represent projective points by nonzero tuples considered equivalent under multiplication by any nonzero scalar. They encode affine points and points at infinity in one system.

Projective plane: The geometric plane whose points are lines through the origin in three-dimensional space. Each projective point is represented by a scale-equivalent triple of homogeneous coordinates.
Vector space: A set whose elements can be added and multiplied by scalars while satisfying the vector-space axioms. Homogeneous tuples are nonzero vectors, and their scalar multiples encode one projective point.
Perspective projection: A mapping that projects three-dimensional points onto an image plane from a viewpoint. Camera projection becomes a matrix multiplication followed by division by the resulting scale coordinate.
Cartesian coordinates: Coordinates that locate points using signed distances along perpendicular axes. Cartesian coordinates describe finite points directly but do not represent directions at infinity as points.
Point at infinity: A projective point representing a direction of parallel affine lines. Triples with last coordinate zero encode directions rather than finite affine points.
Equivalence relation: A relation that is reflexive, symmetric, and transitive, partitioning a set into equivalence classes. Projective coordinates identify all nonzero scalar multiples as equivalent representations.
Camera matrix: A matrix mapping three-dimensional scene coordinates to two-dimensional image coordinates in a camera model. It acts on homogeneous scene points and produces homogeneous image points.
Barycentric coordinates: Coordinates expressing a point as an affine combination of the vertices of a simplex. Barycentric coordinates depend on a chosen simplex, while homogeneous coordinates describe projective incidence.
Affine chart: A coordinate region of projective space obtained by excluding a chosen hyperplane. Setting the final coordinate to one recovers familiar affine coordinates in this chart.
Quotient space: A space formed by treating elements related by an equivalence relation as a single element. Projective space is built from nonzero vectors after identifying vectors on the same line.

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