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In geometry, in the context of a real geometric space extended to (or embedded in) a complex projective space, an imaginary point is a point not contained in the embedded space.


In terms of homogeneous coordinates, a point of the complex projective plane with coordinates (a,b,c) in the complex projective space for which there exists no complex number z such that za, zb, and zc are all real.

This definition generalizes to complex projective spaces. The point with coordinates


is imaginary if there exists no complex number z such that


are all real coordinates.[1]

Every imaginary point belongs to exactly one real line, the line through the point and its complex conjugate.[1]
See also

Real point


Pottmann, Helmut; Wallner, Johannes (2009), Computational Line Geometry, Mathematics and visualization, Springer, pp. 54–55, ISBN 9783642040184.

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