Point at infinity

# Point at infinity

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The point at infinity, also called ideal point, is a point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

which when added to the real number line
Number line
In basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point. Often the integers are shown as specially-marked points evenly spaced on the line...

yields a closed curve called the real projective line, . The real projective line is not equivalent to the extended real number line
Extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...

, which has two different points at infinity.

The point at infinity can also be added to the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

, , thereby turning it into a closed surface (i.e., complex algebraic curve) known as the complex projective line, , also called the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...

.

## Projective planes

Now consider a pair of parallel lines in an affine plane A. Since the lines are parallel
Parallel (geometry)
Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...

, they do not intersect in A, but can be made to intersect in the projective completion of A, a projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...

P, by adding the same point at infinity to each of the lines. In fact, this point at infinity must be added to all of the lines in the parallel class of lines that contains these two lines. Different parallel classes of lines of A will receive different points at infinity. The collection of all the points at infinity form the line at infinity
Line at infinity
In geometry and topology, the line at infinity is a line that is added to the real plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line.-Geometric formulation:In...

. This line at infinity lies in P but not in A. Lines of A which meet in A will get different ideal points since they can not be in the same parallel class, while lines of A which are parallel will get the same ideal point.

In hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

, an ideal point is also called an omega point. Given a line l and a point P not on l, right- and left-limiting parallels
Limiting parallel
In neutral geometry, there may be many lines parallel to a given line l at a point P, however one parallel may be closer to l than all others. Thus it is useful to make a new definition concerning parallels in neutral geometry...

to l through P are said to meet l at omega points (this is an abuse of language since these lines do not meet in the geometry). In the Poincaré disk model
Poincaré disk model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or unit ball, and the straight lines of the hyperbolic geometry are segments of circles contained in the disk...

and the Klein model
Klein model
In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of n-dimensional hyperbolic geometry in which points are represented by the points in the interior of the n-dimensional unit ball and lines are represented by the...

of hyperbolic geometry, the omega points can be visualized since they lie on the boundary circle (which is not part of the model). Pasch's axiom
Pasch's axiom
In geometry, Pasch's axiom is a result of plane geometry used by Euclid, but yet which cannot be derived from Euclid's postulates. Its axiomatic role was discovered by Moritz Pasch.The axiom states that, in the plane,...

and the exterior angle theorem
Exterior angle theorem
The exterior angle theorem can mean one of two things: Postulate 1.16 in Euclid's Elements which states that the exterior angle of a triangle is bigger than either of the remote interior angles, or a theorem in elementary geometry which states that the exterior angle of a triangle is equal to the...

still hold for an omega triangle, defined by two points in hyperbolic space and an omega point.

## Generalisations

This construction can be generalized to an arbitrary topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

. 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.

• Division by zero
Division by zero
In mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...

• Line at infinity
Line at infinity
In geometry and topology, the line at infinity is a line that is added to the real plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line.-Geometric formulation:In...

• Plane at infinity
Plane at infinity
In projective geometry, the plane at infinity is a projective plane which is added to the affine 3-space in order to give it closure of incidence properties. The result of the addition is the projective 3-space, P^3...

• Hyperplane at infinity