Point at infinity
Concept in geometry / From Wikipedia, the free encyclopedia
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
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In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a projective plane, in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over any field, and more generally over any division ring.[1]
In the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the complex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, CP1, also called the Riemann sphere (when complex numbers are mapped to each point).
In the case of a hyperbolic space, each line has two distinct ideal points. Here, the set of ideal points takes the form of a quadric.