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]
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.