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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.)

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