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, mathbb{R}P^1. Nota Bene: The real projective line is not equivalent to the extended real number line.

Related Topics:
Point at infinity - Point - Number line - Closed curve - Projective line - Nota Bene - Extended real number line

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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.)

Related Topics:
Complex plane - Riemann sphere - Sphere

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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.

Related Topics:
Projective plane - Parallel - Line at infinity

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

Related Topics:
Line at infinity - Plane at infinity - Hyperplane at infinity

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