Complex conjugate line

In complex geometry, the complex conjugate line of a straight line is the line that it becomes by taking the complex conjugate of each point on this line.[1]

This is the same as taking the complex conjugates of the coefficients of this line. So if the equation of D is D : ax + by + cz = 0, then the equation of its conjugate D* is D* : a*x + b*y + c*z = 0.

The conjugate of a real line is the line itself. The intersection point of two conjugated lines is always real.[2]

References

Shafarevich, Igor R.; Remizov, Alexey; Kramer, David P.; Nekludova, Lena (2012), Linear Algebra and Geometry, Springer, p. 413, ISBN 9783642309946.
Schwartz, Laurent (2001), A Mathematician Grappling With His Century, Springer, p. 52, ISBN 9783764360528.

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[1] Shafarevich, Igor R.; Remizov, Alexey; Kramer, David P.; Nekludova, Lena (2012), Linear Algebra and Geometry, Springer, p. 413, ISBN 9783642309946.

[2] Schwartz, Laurent (2001), A Mathematician Grappling With His Century, Springer, p. 52, ISBN 9783764360528.