III.72 Projective Space

The real projective plane can be defined in various ways. One way is to use three homogeneous coordinates: a typical point is represented as (x, y, z), where not all of x, y, and z are equal to 0, with the convention that if is a nonzero constant, then (x, y, z) and (x, y, z) are regarded as equal. Notice that for each (x, y, z) the set of all points of the form (x, y, z) is the line through the origin and (x, y, z), and indeed a more geometrical definition of the real projective plane is that it is the set of all lines in ³ that pass through the origin. Each such line meets the unit sphere in exactly two points, which are opposite each other, and a third way of defining the real projective plane is to define opposite points in the unit sphere to be equivalent and to take the QUOTIENT [I.3 §3.3] Of the unit sphere by this EQUIVALENCE RELATION [I.2 §2.3]. A fourth way to define the projective plane is to start with the usual Euclidean plane and to add one “point at infinity” for each possible slope that a line can have. With an appropriate topology, this defines the projective plane as a COMPACTIFICATION [III.9] of the Euclidean plane.

Taking the third definition, a line in the projective plane is defined to be a great circle with its opposite points identified. It is then not hard to see that any two lines meet in exactly one point (since any two great circles meet in exactly two opposite points) and that any two points are contained in exactly one line. This property can be used to define much more abstract generalizations of the notion of a projective plane.

There are similar definitions for other fields besides , and also in higher dimensions. For instance, complex projective n-space is the set of all points of the form (z_l, z₂, . . . , Z_n+1), where not every z_i is 0, with (z_l, z₂, . . . , z_n+1) equivalent to (z₁, z₂, . . . , z_n+1) if is a nonzero complex scalar. This is the set of all “complex lines” in ^n+l that pass through the origin. See SOME FUNDAMENTAL MATHEMATICAL DEFINITIONS [I.3 §6.7] for more details about projective geometry.