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]