# 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, projective planes, plane":

Spherical Polyhedron - Relation To Tilings of The Projective Plane
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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 ...
Lift (mathematics)
... on a sphere 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 ... on the sphere, the lift of the path in the projective plane ...
Examples - Extremal Length of Topologically Essential Paths in Projective Plane
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Conic Section - Pencil of Conics
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### Famous quotes containing the word plane:

It was the most ungrateful and unjust act ever perpetrated by a republic upon a class of citizens who had worked and sacrificed and suffered as did the women of this nation in the struggle of the Civil War only to be rewarded at its close by such unspeakable degradation as to be reduced to the plane of subjects to enfranchised slaves.
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