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Concept# Point at infinity

Summary

In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
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.
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.
In an affine or Euclidean space of higher dimension, the points at infinity are the points which are added to the space to get the projective completion. The set of the points at infinity is called, depending on the dimension of the space, the line at infinity, the plane at infinity or the hyperplane at infinity, in all cases a projective space of one less dimension.
As a projective space over a field is a smooth algebraic variety, the same is true for the set of points at infinity. Similarly, if the ground field is the real or the complex field, the set of points at infinity is a manifold.
Perspective (graphical)
In artistic drawing and technical perspective, the projection on the picture plane of the point at infinity of a class of parallel lines is called their vanishing point.
Ideal point
In hyperbolic geometry, points at infinity are typically named ideal points.

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Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value for infinity. With the Riemann model, the point is near to very large numbers, just as the point is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as well-behaved.

Homography

In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation.

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

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