a point at infinity

An artist drawing a tiled floor in perspective lets the receding edges of the tiles converge to a single dot on the horizon, the 'vanishing point'. That dot is not a real place in the room — no tile reaches it — yet the drawing treats it as a perfectly good point where the parallel edges meet. A point at infinity is exactly this idea made precise: a single new point attached to each direction of the plane, the place where all lines of that direction are declared to meet.

Here is the precise rule. Two lines in the Euclidean plane have the same direction (are parallel) exactly when they have the same slope. Group the lines by slope; to each slope-class attach one new point, its point at infinity, and decree that every line in the class passes through that one point. So all lines of slope 2 now meet at the slope-2 point at infinity; all vertical lines meet at the vertical point at infinity. A line that already crossed another line keeps its ordinary intersection; lines that used to be parallel now intersect at their shared infinity-point. In homogeneous coordinates (x : y : z) a point at infinity is one with z = 0, for instance (1 : 2 : 0) is the point at infinity in the direction of slope 2.

The payoff is that the annoying exception 'two lines meet in one point, unless they are parallel' disappears: with infinity-points added, any two distinct lines meet in exactly one point, full stop. A caution worth stating plainly: a line and the line obtained by reversing its arrow point to the same point at infinity — direction here means a slope, an unsigned orientation, not a one-way arrow. So a line has just one point at infinity, not two.