# 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, plane, projective planes, projective":

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 ...
Conic Section - Pencil of Conics
... by 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 ... 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 four points ... Let C1 and C2 be two distinct conics in a projective plane defined over an algebraically closed field K ...
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 geometry ...
Spherical Polyhedron - Relation To Tilings of The Projective Plane
... 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 covering ... 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 ... path on the sphere, the lift of the path in the projective plane ...

### Famous quotes containing the word plane:

Even though I had let them choose their own socks since babyhood, I was only beginning to learn to trust their adult judgment.. . . I had a sensation very much like the moment in an airplane when you realize that even if you stop holding the plane up by gripping the arms of your seat until your knuckles show white, the plane will stay up by itself. . . . To detach myself from my children . . . I had to achieve a condition which might be called loving objectivity.
—Anonymous Parent of Adult Children. Ourselves and Our Children, by Boston Women’s Health Book Collective, ch. 5 (1978)