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In complex geometry, an **imaginary line** is a straight line that only contains one real point. It can be proven that this point is the intersection point with the conjugated line.^{[1]}

It is a special case of an imaginary curve.

An imaginary line is found in the complex projective plane P^{2}(C) where points are represented by three homogeneous coordinates

Boyd Patterson described the lines in this plane:^{[2]}

- The locus of points whose coordinates satisfy a homogeneous linear equation with complex coefficients
- is a straight line and the line is
*real*or*imaginary*according as the coefficients of its equation are or are not proportional to three real numbers.

Felix Klein described imaginary geometrical structures: "We will characterize a geometric structure as imaginary if its coordinates are not all real.:^{[3]}

According to Hatton:^{[4]}

- The locus of the double points (imaginary) of the overlapping involutions in which an overlapping involution pencil (real) is cut by real transversals is a pair of imaginary straight lines.

Hatton continues,

- Hence it follows that an imaginary straight line is determined by an imaginary point, which is a double point of an involution, and a real point, the vertex of the involution pencil.

- J.L.S. Hatton (1920) The Theory of the Imaginary in Geometry together with the Trigonometry of the Imaginary, Cambridge University Press via Internet Archive
- Felix Klein (1928)
*Vorlesungen über nicht-euklischen Geometrie*, Julius Springer.