# Projective Plane

A projective plane consists of a set of lines, a set of points, and a relation between points and lines called incidence, having the following properties:

1. Given any two distinct points, there is exactly one line incident with both of them.
2. Given any two distinct lines, there is exactly one point incident with both of them.
3. There are four points such that no line is incident with more than two of them.

The second condition means that there are no parallel lines. The last condition excludes the so-called degenerate cases (see below). The term "incidence" is used to emphasize the symmetric nature of the relationship between points and lines. Thus the expression "point P is incident with line l " is used instead of either "P is on l " or "l passes through P ".

### Other articles related to "projective plane, projective planes, projective, plane":

Examples - Extremal Length of Topologically Essential Paths in Projective Plane
... The projective plane with the spherical metric is obtained by identifying antipodal points on the unit sphere in with its Riemannian spherical metric ... Let denote the set of closed curves in this projective plane that are not null-homotopic ...
Projective Planes in Higher Dimensional Projective Spaces
... Projective planes may be thought of as projective geometries of "geometric" dimension two ... Higher dimensional projective geometries can be defined in terms of incidence relations in a manner analogous to the definition of a projective plane ... These turn out to be "tamer" than the projective planes since the extra degrees of freedom permit Desargues' theorem to be proved geometrically in the higher dimensional ...
Spherical Polyhedron - Relation To Tilings of The Projective Plane
... Spherical polyhedra having at least one inversive symmetry are related to projective polyhedra (tessellations of the real projective plane) – just as the sphere has a 2-to-1 ... For example, the 2-fold cover of the (projective) hemi-cube is the (spherical) cube ...
Lift (mathematics)
... to the same point, a continuous map from the sphere covering the projective plane ... A path in the projective plane is a continuous map from the unit interval ... the sphere, the lift of the path in the projective plane ...
Conic Section - Pencil of Conics
... five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics ... In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these ... Let C1 and C2 be two distinct conics in a projective plane defined over an algebraically closed field K ...

### Famous quotes containing the word plane:

We’ve got to figure these things a little bit different than most people. Y’know, there’s something about going out in a plane that beats any other way.... A guy that washes out at the controls of his own ship, well, he goes down doing the thing that he loved the best. It seems to me that that’s a very special way to die.
Dalton Trumbo (1905–1976)