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In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space ''V''. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers. The idea of a projective space relates to perspective, more precisely to the way an eye or a camera projects a 3D scene to a 2D image. All points that lie on a projection line (i.e., a "line-of-sight"), intersecting with the entrance pupil of the camera, are projected onto a common image point. In this case, the vector space is R3 with the camera entrance pupil at the origin, and the projective space corresponds to the image points. Projective spaces can be studied as a separate field in mathematics, but are also used in various applied fields, geometry in particular. Geometric objects, such as points, lines, or planes, can be given a representation as elements in projective spaces based on homogeneous coordinates. As a result, various relations between these objects can be described in a simpler way than is possible without homogeneous coordinates. Furthermore, various statements in geometry can be made more consistent and without exceptions. For example, in the standard Euclidian geometry for the plane, two lines always intersect at a point ''except'' when the lines are parallel. In a projective representation of lines and points, however, such an intersection point exists even for parallel lines, and it can be computed in the same way as other intersection points. Other mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, and their representation theories. ==Introduction== As outlined above, projective space is a geometric object that formalizes statements like "Parallel lines intersect at infinity." For concreteness, we give the construction of the real projective plane P2(R) in some detail. There are three equivalent definitions: #The set of all lines in R3 passing through the origin (0, 0, 0). Every such line meets the sphere of radius one centered in the origin exactly twice, say in and its antipodal point . #P2(R) can also be described as the points on the sphere ''S''2, where every point ''P'' and its antipodal point are not distinguished. For example, the point (red point in the image) is identified with (light red point), etc. #Finally, yet another equivalent definition is the set of equivalence classes of , i.e., 3-space without the origin, where two points and are equivalent iff there is a nonzero real number ''λ'' such that , i.e., , , . The usual way to write an element of the projective plane, i.e., the equivalence class corresponding to an honest point in R3, is . The last formula goes under the name of homogeneous coordinates. In homogeneous coordinates, any point with is equivalent to . So there are two disjoint subsets of the projective plane: that consisting of the points for , and that consisting of the remaining points . The latter set can be subdivided similarly into two disjoint subsets, with points and . In the last case, ''x'' is necessarily nonzero, because the origin was not part of P2(R). This last point is equivalent to . Geometrically, the first subset, which is isomorphic (not only as a set, but also as a manifold, as seen later) to R2, is in the image the yellow upper hemisphere (without the equator), or equivalently the lower hemisphere. The second subset, isomorphic to R1, corresponds to the green line (without the two marked points), or, again, equivalently the light green line. Finally we have the red point or the equivalent light red point. We thus have a disjoint decomposition :P2(R) = R2 ⊔ R1 ⊔ ''point''. Intuitively, and made precise below, R1 ⊔ ''point'' is itself the real projective line P1(R). Considered as a subset of P2(R), it is called ''line at infinity'', whereas is called ''affine plane'', i.e., just the usual plane. The next objective is to make the saying "parallel lines meet at infinity" precise. A natural bijection between the plane (which meets the sphere at the north pole and the sphere of the projective plane is accomplished by the gnomonic projection. Each point ''P'' on this plane is mapped to the two intersection points of the sphere with the line through its center and ''P''. These two points are identified in the projective plane. Lines (blue) in the plane are mapped to great circles if one also includes one pair of antipodal points on the equator. Any two great circles intersect precisely in two antipodal points (identified in the projective plane). Great circles corresponding to parallel lines intersect on the equator. So ''any'' two lines have exactly one intersection point inside P2(R). This phenomenon is axiomatized in projective geometry. 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 space」の詳細全文を読む スポンサード リンク
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