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projective space[prə′jek·tiv ′spās]
in its original meaning, Euclidean space with the addition of the plane at infinity. Points at infinity, lines at infinity, and the plane at infinity are called ideal elements. In projective space, one ideal point is added to each line, one ideal line is added to each plane, and one ideal plane is added to the entire space. Parallel lines are supplemented by a common ideal point, and nonparallel lines by distinct ideal points. Parallel planes are supplemented by a common ideal line, and nonparallel planes by distinct ideal lines. The ideal points added to all the lines of a given plane belong to the ideal line added to this plane. All the ideal points and lines belong to the ideal plane.
Projective space can be analytically defined as the set of classes of proportional quadruples of real numbers that do not simultaneously equal zero. The classes can be interpreted as the points of projective space, in which case the numbers in the quadruples are called the homogeneous coordinates of the points. The classes can also be viewed as the planes of the projective space, in which case the numbers are called the homogeneous coordinates of the planes. The relation of incidence for a point (x1: x2: x3: x4) and a plane (u1: u2: u3: u4) is expressed by the equation
The concept of an n-dimensional projective space, which plays an important role in algebraic geometry, can be introduced in an analogous fashion. The coordinates of such a space are the elements of some skew field k. In the most general sense, a projective space is a collection of three sets of elements, called points, lines, and planes, for which the relations of membership and order are defined in such a way that the axioms of projective geometry are satisfied. A. N. Kolmogorov and L. S. Pontriagin have proved that if a projective space over a skew field k is a connected compact topological space, in which a line depends continuously on two of its points, and if the incidence axioms are satisfied, then k is the field of real numbers, the field of complex numbers, or the skew field of quaternions.