Projective plane | Wikipedia audio article
Channel: wikipedia tts
Date: 29/05/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Projective_plane


00:01:41 1 Definition
00:02:31 2 Some examples
00:02:41 2.1 The extended Euclidean plane
00:03:51 2.2 Projective Moulton plane
00:05:01 2.3 A finite example
00:06:35 3 Vector space construction
00:08:22 3.1 Classical examples
00:11:12 3.2 Finite field planes
00:12:45 3.3 Desargues' theorem and Desarguesian planes
00:14:49 4 Subplanes
00:14:57 4.1 Fano subplanes
00:15:09 5 Affine planes
00:16:08 5.1 Definition
00:17:37 5.2 Construction of projective planes from affine planes
00:17:49 5.3 Generalized coordinates
00:20:03 6 Degenerate planes
00:20:30 7 Collineations
00:22:15 7.1 Homography
00:28:58 8 Plane duality
00:30:17 9 Correlations
00:32:39 10 Finite projective planes
00:33:03 11 Projective planes in higher-dimensional projective spaces
00:33:56 12 See also
00:34:29 13 Notes
00:35:06 14 References
00:37:22 15 External links
00:37:27 Collineations
00:37:37 A collineation of a projective plane is a bijective map of the plane to itself which maps points to points and lines to lines that preserves incidence, meaning that if σ is a bijection and point P is on line m, then Pσ is on mσ.If σ is a collineation of a projective plane, a point P with P
00:38:07 mσ is called a fixed line of σ. The points on a fixed line need not be fixed points, their images under σ are just constrained to lie on this line. The collection of fixed points and fixed lines of a collineation form a closed configuration, which is a system of points and lines that satisfy the first two but not necessarily the third condition in the definition of a projective plane. Thus, the fixed point and fixed line structure for any collineation either form a projective plane by themselves, or a degenerate plane. Collineations whose fixed structure forms a plane are called planar collineations.
00:38:54 Homography
00:39:03 A homography (or projective transformation) of PG(2,K) is a collineation of this type of projective plane which is a linear transformation of the underlying vector space.  Using homogeneous coordinates they can be represented by invertible 3 × 3 matrices over K which act on the points of PG(2,K) by y
00:40:40 Plane duality
00:43:22 Correlations
00:43:35 (P, L, I) to its dual plane C*
00:44:01 L and Lσ
00:45:18 Finite projective planes
00:46:48 6.  The next case N
00:49:12 Projective planes in higher-dimensional projective spaces
00:51:27 See also



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SUMMARY
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In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect in one and only one point.
Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG(2, R),  RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane.
A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional projective spaces. Such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes.