Projective plane

A projective plane extends the Euclidean plane with ideal points that represent directions, so every pair of distinct lines intersects at exactly one point.

Projective geometry: The geometry of properties preserved by projective transformations, especially incidence and cross-ratio. The projective plane is a fundamental space studied by this geometry.
Real projective plane: The projective plane over the real numbers, commonly represented by nonzero triples modulo scaling. It supplies the standard coordinate model for the plane described here.
Complex projective plane: The projective plane over the complex numbers, whose points are nonzero complex triples modulo scaling. It is a major alternative coordinate model to the real projective plane.
Girard Desargues: A seventeenth-century French mathematician whose work helped establish projective geometry. His theorem became a key structural property of many projective planes.
Point at infinity: A point added to an affine space to represent a direction of parallel lines. Each parallel class of lines meets at its corresponding ideal point.
Affine plane: A plane geometry in which parallel lines remain disjoint and points have vector coordinates. Adding its direction classes as ideal points produces a projective completion.
Finite projective plane: A projective plane with finitely many points and lines, satisfying the projective incidence axioms. It shows the same incidence principles can hold in a finite space.
Jean-Victor Poncelet: A French mathematician who developed projective geometry systematically in the early nineteenth century. His work advanced the geometric treatment of ideal points and projection.
Line at infinity: The line consisting of all points at infinity in a projective completion of an affine plane. It gathers the plane’s ideal points into one projective line.
Projective transformation: A transformation induced by an invertible linear map on homogeneous coordinates, defined up to scalar. It preserves incidence while moving points and lines across the plane.

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