In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane.
Therefore, lines in the projective plane are closed curves, i.e., they are cyclical rather than linear.
Likewise, a parabola can be seen as a closed curve which intersects the line at infinity in a single point.
In fact one of the most applied tricks was to regard a circle as a conic constrained to pass through two points at infinity, the solutions of This equation is the form taken by that of any circle when we drop terms of lower order in X and Y.
More formally, we should use homogeneous coordinates and note that the line at infinity is specified by setting Making equations homogeneous by introducing powers of Z, and then setting Z = 0, does precisely eliminate terms of lower order.
The conclusion is that the three-parameter family of circles can be treated as a special case of the linear system of conics passing through two given distinct points P and Q.