# Point at infinity

The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, [itex]\mathbb{R}P^1[itex]. Nota Bene: The real projective line is not equivalent to the extended real number line.

The point at infinity can also be added to the complex plane, [itex]\mathbb{C}^1[itex], thereby turning it into a closed surface known as the complex projective line, [itex]\mathbb{C}P^1[itex], a.k.a. Riemann sphere. (A sphere with a hole punched into it and its resulting edge being pulled out towards infinity is a plane. The reverse process turns the complex plane into [itex]\mathbb{C}P^1[itex]: the hole is un-punched by adding a point to it which is identically equivalent to each and every one of the points on the rim of the hole.)

Now consider a pair of parallel lines in a projective plane [itex]\mathbb{R}P^2[itex]. Since the lines are parallel, they intersect at a point at infinity which lies on [itex]\mathbb{R}P^2[itex]'s line at infinity. Moreover, each of the two lines is, in [itex]\mathbb{R}P^2[itex], a projective line: each one has its own point at infinity. When a pair of projective lines are parallel they intersect at their common point at infinity.

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