Point At Infinity

(Hyperbolic Geometry)

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Hyperbolic Geometry
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What is Point at infinity?

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.

Technology Types

hyperbolic geometryinfinityprojective geometry

Synonyms

Improper pointPoints at infinityProjective infinityUnsigned infinity

Translations

Enti geometrici impropri (it)Fernelement (de)Nevlastní bod (cs)Point à l'infini (fr)Ponto impróprio (pt)Punt de l'infinit (ca)Punt op oneindig (nl)Punto del infinito (es)Бесконечно удалённая точка (ru)Точка на нескінченності (uk)كيانات هندسية لانهائية (ar)무한원점 (ko)无穷远点 (zh)無限遠点 (ja)

Tech Info


Source: [object Object]
 — Date merged: 11/6/2021, 1:32:41 PM
 — Date scraped: 5/20/2021, 5:42:10 PM