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point at issue

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. The real projective line is not equivalent to the extended real number line, which has two different points at infinity.

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, also called the 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.)

This construction can be generalized to an arbitrary topological space. The space so obtained is called the one-point compactification or Alexandroff compactification of the original space. Thus the circle is the one-point compactification of the line, and the sphere is the one-point compactification of the plane.

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.

In hyperbolic geometry, the ideal point is also called the omega point. Given a line l and a point P not on l, right- and left-limiting parallels to l through P meet at a point on the boundary circle of the Poincaré disk model and the Klein model called the omega point. Pasch's axiom and the Exterior Angle Theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point.

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