Point at infinity

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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, \mathbb{R}P^1. 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, \mathbb{C}^1, thereby turning it into a closed surface known as the complex projective line, \mathbb{C}P^1, 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 \mathbb{C}P^1: the hole is un-punched by adding a point to it which is identically equivalent to each of the points on the rim of the hole.)

Now consider a pair of parallel lines in a projective plane \mathbb{R}P^2. Since the lines are parallel, they intersect at a point at infinity which lies on \mathbb{R}P^2's line at infinity. Moreover, each of the two lines is, in \mathbb{R}P^2, 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.

See also: line at infinity, plane at infinity, hyperplane at infinity

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