complex conjugate line

{{short description|Operation in complex geometry}}

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.{{citation|title=Linear Algebra and Geometry|first1=Igor R.|last1=Shafarevich|first2=Alexey|last2=Remizov|first3=David P.|last3=Kramer|first4=Lena|last4=Nekludova|publisher=Springer|year=2012|isbn=9783642309946|page=413|url=https://books.google.com/books?id=6Pp2-DTOKWIC&pg=PA413}}.

This is the same as taking the complex conjugates of the coefficients of the line. So if the equation of {{math|D}} is {{math|1=D: ax + by + cz = 0}}, then the equation of its conjugate {{math|D*}} is {{math|1=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.{{citation|title=A Mathematician Grappling With His Century|first=Laurent|last=Schwartz|publisher=Springer|year=2001|isbn=9783764360528|page=52|url=https://books.google.com/books?id=xVrC7ek7iRwC&pg=PA52}}.