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 the 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.

[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.

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