Imaginary point

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.

Definition

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

$(a_1,a_2,\ldots,a_n)$

is imaginary if there exists no complex number z such that

$(za_1,za_2,\ldots,za_n)$

are all real coordinates.^[1]

Properties

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

See also

• Real point

References

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