How locus of points of parallel lines in homogeneous coordinates, forms infinity?

Question

In the diagram shown in link, what does writer mean when he says "locus of these points forms the line r". In the diagram "r" is curved, why is it called a line. I am facing difficulty in grabbing the concept of infinity in homogeneous coordinates, like, how does [wa,wb,0] define infinity. Can you please depict this visually?

Projective geometry

Answer 1

Your blue annotation is misleading. All the parallelel lines are considered to converge to a single point at infinity.

Consider a pencil of converging lines, and let the intersection point escape to infinity in some direction: the lines become parallel. Now rotate the figure; you can imagine that the meeting point describes a circle of infinite radius. The latter is called the line at infinity. Its exact shape is immaterial, though you can say that a circle of infinite radius has zero curvature, i.e. is straight everywhere.