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, . 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, , thereby turning it into a closed surface known as the complex projective line, , 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 : 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 . Since the lines are parallel, they intersect at a point at infinity which lies on 's line at infinity. Moreover, each of the two lines is, in , 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 |