Abstract
A straight line intersects a circle in two, one, or no real points. In the last case, they have two complex conjugate intersecting points.
We present their construction by tracing the circle with all lines. To visualize these points, the real plane is extended with the imaginary dimensions to four-dimensional real space.
The surface generated by all complex points is orthogonally projected into two three-dimensional subspaces generated by both real and one of the imaginary dimensions. The same method is used to trace and visualize other real and imaginary conics and a cubic curve.
Furthermore, we describe a graphical representation of complex lines in the four-dimensional space and discuss the elementary incidence properties of points and lines. This paper provides an accessible method of visualization of the complex number plane.