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Concept# Real point

Summary

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:
: (u_1, u_2, \ldots, u_n)
are the homogeneous coordinates of a real point if there exists a nonzero complex number λ such that the coordinates of
: (\lambda u_1, \lambda u_2, \ldots, \lambda u_n)
are all real.
A point which is not real is called an imaginary point.
Context
Geometries that are specializations of real projective geometry, such as Euclidean geometry, elliptic geometry or conformal geometry may be complexified, thus embedding the points of the geometry in a complex projective space, but retaining the identity of the original real space as special. Lines, planes etc. are expanded to the lines, etc. of the complex projective spac

Official source

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