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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 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 , 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. The embedding property is a consequence of a result known as Desargues' theorem. ==Definition== 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:〔In a more formal version of the definition it is pointed out that the terms ''point, line'' and ''incidence'' are primitive notions (undefined terms). This formal viewpoint is needed to understand the concept of duality when applied to projective planes.〕 #Given any two distinct points, there is exactly one line incident with both of them. #Given any two distinct lines, there is exactly one point incident with both of them. #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'' ". 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Projective plane」の詳細全文を読む スポンサード リンク
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