In geometry, a real point is a point in the complex projective plane with homogeneous coordinates (x,y,z) for which there exists a nonzero complex number λ such that λx, λy, and λz are all real numbers.
This definition can be widened to a complex projective space of arbitrary finite dimension as follows: are the homogeneous coordinates of a real point if there exists a nonzero complex number λ such that the coordinates of are all real.
As with the inclusion of points at infinity and complexification of real polynomials, this allows some theorems to be stated more simply without exceptions and for a more regular algebraic analysis of the geometry.
Viewed in terms of homogeneous coordinates, a real vector space of homogeneous coordinates of the original geometry is complexified.
A point of the original geometric space is defined by an equivalence class of homogeneous vectors of the form λu, where λ is an nonzero complex value and u is a real vector.