Point at infinity

The real line with the point at infinity.

The point at infinity, also called the ideal point, of the real number line is a point which, when added to the number line yields a closed curve called the real projective line, $\mathbb{R}P^1$ . 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, $\mathbb{C}^1$ , thereby turning it into a closed surface (i.e., complex algebraic curve) known as the complex projective line, $\mathbb{C}P^1$ , also called the Riemann sphere.

The concept of infinity point admits several generalizations for various multi-dimensional constructions.

Projective geometry

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.

This condition does not depend on the ground field. If real or complex numbers are used, then, from the point of view of differential geometry, points at infinity form a hypersurface, which means a submanifold having one less dimension than the whole projective space. In the general case these facts may be formulated using algebraic manifolds.

Projective plane

Consider a pair of parallel lines in an affine plane A. Since the lines are parallel, they do not intersect in A, but can be made to intersect in the projective completion of A, a projective plane 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. 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.

The line at infinity is itself a projective line over the same ground field. For example, it is topologically a circle for the real projective plane, and a sphere for the complex projective plane.

Perspective

Main article: 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.

Hyperbolic geometry

In hyperbolic geometry, 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 to l through P are said to meet l at omega points. Unlike the projective case, omega points form a boundary, not a submanifold. So, these lines do not intersect at an omega point and such points, although well defined, do not belong to a hyperbolic space itself.

The points at infinity form the Cayley absolute of a non-Euclidean geometry. For instance, the unit circle forms the Cayley absolute of the Poincaré disk model and the real line forms the Cayley absolute of Poincaré half-plane model of hyperbolic geometry.^[1]

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.^[2]

Other generalisations

Main article: Compactification (mathematics)

This construction can be generalized to topological spaces. Different compactifications may exist for a given space, but arbitrary topological space admits Alexandroff extension, also called the one-point compactification when the original space is not itself compact. Projective line (over arbitrary field) is the Alexandroff extension of the corresponding field. Thus the circle is the one-point compactification of the real line, and the sphere is the one-point compactification of the plane. Projective spaces P^$n$ for $n$ > 1 are not one-point compactifications of corresponding affine spaces for the reason mentioned above, and completions of hyperbolic spaces with omega points are also not one-point compactifications.

References

↑ Struve, Horst; Struve, Rolf (2010), "Non-euclidean geometries: the Cayley-Klein approach", Journal of Geometry 89 (1): 151–170, doi:10.1007/s00022-010-0053-z, ISSN 0047-2468, MR 2739193
↑ Hvidsten, Michael (2005). Geometry with Geometry Explorer. New York, NY: McGraw-Hill. pp. 276–283. ISBN 0-07-312990-9.