Imaginary point
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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.
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
- [math]\displaystyle{ (a_1,a_2,\ldots,a_n) }[/math]
is imaginary if there exists no complex number z such that
- [math]\displaystyle{ (za_1,za_2,\ldots,za_n) }[/math]
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
References
- ↑ 1.0 1.1 Pottmann, Helmut; Wallner, Johannes (2009), Computational Line Geometry, Mathematics and visualization, Springer, pp. 54–55, ISBN 9783642040184, https://books.google.com/books?id=LadAAAAAQBAJ&pg=PA5.
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