# 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, $mathbb\left\{R\right\}P^1$. 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, $mathbb\left\{C\right\}^1$, thereby turning it into a closed surface known as the complex projective line, $mathbb\left\{C\right\}P^1$, 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 $mathbb\left\{C\right\}P^1$: 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 $mathbb\left\{R\right\}P^2$. Since the lines are parallel, they intersect at a point at infinity which lies on $mathbb\left\{R\right\}P^2$'s line at infinity. Moreover, each of the two lines is, in $mathbb\left\{R\right\}P^2$, 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. [cite book|last =Hvidsten|first =Michael|title = Geometry with Geometry Explorer|publisher = McGraw-Hill|year = 2005 | location = New York, NY | pages = 276-283 | isbn = 0-07-312990-9]

See also

*line at infinity
*plane at infinity
*hyperplane at infinity

References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• point at infinity — an ideal mathematical point in projective geometry that preserves the magnitudes of all angles of a transformed plane …   Useful english dictionary

• Infinity — In mathematics, infinity is often used in contexts where it is treated as if it were a number (i.e., it counts or measures things: an infinite number of terms ) but it is a different type of number from the real numbers. Infinity is related to… …   Wikipedia

• infinity — noun 1 endless space/time VERB + INFINITY ▪ extend into, extend to, stretch into, stretch to ▪ Theoretically, a line can extend into infinity. PREPOSITION ▪ at …   Collocations dictionary

• Infinity — • The infinite, as the word indicates, is that which has no end, no limit, no boundary, and therefore cannot be measured by a finite standard, however often applied; it is that which cannot be attained by successive addition, not exhausted by… …   Catholic encyclopedia

• Infinity (wargame) — Infinity is a science fiction skirmish level tabletop miniature wargame produced by Corvus Belli. Set about 175 years into the future, players of the game enact a skirmish between opposing forces using 28 mm scale miniatures and utilizing a… …   Wikipedia

• infinity — [in fin′i tē] n. pl. infinities [ME infinite < OFr infinité < L infinitas] 1. the quality of being infinite 2. anything infinite; endless or unlimited space, time, distance, quantity, etc. 3. an indefinitely large number or amount 4. Geom.… …   English World dictionary

• infinity — ► NOUN (pl. infinities) 1) the state or quality of being infinite. 2) a very great number or amount. 3) Mathematics a number greater than any assignable quantity or countable number (symbol ). 4) a point in space or time that is or seems… …   English terms dictionary

• infinity — /in fin i tee/, n., pl. infinities. 1. the quality or state of being infinite. 2. something that is infinite. 3. infinite space, time, or quantity. 4. an infinite extent, amount, or number. 5. an indefinitely great amount or number. 6. Math. a.… …   Universalium

• Infinity on High — Infinity on High …   Wikipedia

• Infinity-Man — This article is about the New Gods character; for information on the (unrelated) LSH character with a similar name, please see Infinite Man. Superherobox caption=Infinity Man from Forever People #2, artist Jack Kirby character name=Infinity Man… …   Wikipedia

### Share the article and excerpts

##### Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.