Real linear equation of a line in the complex plane
= Real linear equation of a line in the complex plane
{title2=$\overline c z+c\overline z+r=0$}
For $c\ne0$ and real $r$, this equation defines a <straight line>. Writing $z=x+iy$ and $c=u+iv$ gives $2ux+2vy+r=0$, so $c$ supplies a normal direction. Conversely every real affine line $Ax+By+C=0$ has this representation with $c=(A+iB)/2$ and $r=C$. The condition $c\ne0$ excludes an empty set or the entire <complex plane>.