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, <math>\mathbb{R}P^1<math>. 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, <math>\mathbb{C}^1<math>, thereby turning it into a closed surface known as the complex projective line, <math>\mathbb{C}P^1<math>, 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 <math>\mathbb{C}P^1<math>: 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 <math>\mathbb{R}P^2<math>. Since the lines are parallel, they intersect at a point at infinity which lies on <math>\mathbb{R}P^2<math>'s line at infinity. Moreover, each of the two lines is, in <math>\mathbb{R}P^2<math>, 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 infinityja:無限遠点