Plane at infinity

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In projective geometry, a plane at infinity is the hyperplane at infinity of a three dimensional projective space or to any plane contained in the hyperplane at infinity of any projective space of higher dimension. This article will be concerned solely with the three-dimensional case.

Projective geometry Type of geometry

Projective geometry is a topic in mathematics. It is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points to Euclidean points, and vice versa.

In geometry, any hyperplane H of a projective space P may be taken as a hyperplane at infinity. Then the set complement P ∖ H is called an affine space. For instance, if (x1, ..., xn, xn+1) are homogeneous coordinates for n-dimensional projective space, then the equation xn+1 = 0 defines a hyperplane at infinity for the n-dimensional affine space with coordinates (x1, ..., xn). H is also called the ideal hyperplane.

Projective space space of 1-dimensional linear subspaces (lines passing through the origin) in a vector space

In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when V = R2 and V = R3 are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.

Contents

Definition

There are two approaches to defining the plane at infinity which depend on whether one starts with a projective 3-space or an affine 3-space.

Affine space geometric structure that generalizes the Euclidean space

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

If a projective 3-space is given, the plane at infinity is any distinguished projective plane of the space. [1] This point of view emphasizes the fact that this plane is not geometrically different than any other plane. On the other hand, given an affine 3-space, 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. Meaning that the points of the plane at infinity are the points where parallel lines of the affine 3-space will meet, and the lines are the lines where parallel planes of the affine 3-space will meet. The result of the addition is the projective 3-space, . This point of view emphasizes the internal structure of the plane at infinity, but does make it look "special" in comparison to the other planes of the space.

Projective plane Geometric concept of a 2D space with a "point at infinity" adjoined

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. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect in one and only one point.

In geometry, an incidence relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a point lies on a line" or "a line is contained in a plane" are used. The most basic incidence relation is that between a point, P, and a line, l, sometimes denoted P I l. If P I l the pair (P, l) is called a flag. There are many expressions used in common language to describe incidence but the term "incidence" is preferred because it does not have the additional connotations that these other terms have, and it can be used in a symmetric manner. Statements such as "line l1 intersects line l2" are also statements about incidence relations, but in this case, it is because this is a shorthand way of saying that "there exists a point P that is incident with both line l1 and line l2". When one type of object can be thought of as a set of the other type of object then an incidence relation may be viewed as containment.

Line (geometry) straight object with negligible width and depth

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects. Until the 17th century, lines were defined as the "[…] first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width. […] The straight line is that which is equally extended between its points."

If the affine 3-space is real, , then the addition of a real projective plane at infinity produces the real projective 3-space .

Real projective plane

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R3 passing through the origin.

Analytic representation

Since any two projective planes in a projective 3-space are equivalent, we can choose a homogeneous coordinate system so that any point on the plane at infinity is represented as (X:Y:Z:0). [2] Any point in the affine 3-space will then be represented as (X:Y:Z:1). The points on the plane at infinity seem to have three degrees of freedom, but homogeneous coordinates are equivalent up to any rescaling:

Homogeneous coordinates

In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrischeCalcül, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix.

Up to Mathematical statement of uniqueness, except for an equivalent . structure (equivalence relation)

In mathematics, the phrase up to appears in discussions about the elements of a set, and the conditions under which subsets of those elements may be considered equivalent. The statement "elements a and b of set S are equivalent up to X" means that a and b are equivalent if criterion X is ignored. That is, a and b can be transformed into one another if a transform corresponding to X is applied.

,

so that the coordinates (X:Y:Z:0) can be normalized, thus reducing the degrees of freedom to two (thus, a surface, namely a projective plane).

Proposition: Any line which passes through the origin (0:0:0:1) and through a point (X:Y:Z:1) will intersect the plane at infinity at the point (X:Y:Z:0).

Proof: A line which passes through points (0:0:0:1) and (X:Y:Z:1) will consist of points which are linear combinations of the two given points:

For such a point to lie on the plane at infinity we must have, . So, by choosing , we obtain the point , as required. Q.E.D.

Any pair of parallel lines in 3-space will intersect each other at a point on the plane at infinity. Also, every line in 3-space intersects the plane at infinity at a unique point. This point is determined by the direction—and only by the direction—of the line. To determine this point, consider a line parallel to the given line, but passing through the origin, if the line does not already pass through the origin. Then choose any point, other than the origin, on this second line. If the homogeneous coordinates of this point are (X:Y:Z:1), then the homogeneous coordinates of the point at infinity through which the first and second line both pass is (X:Y:Z:0).

Example: Consider a line passing through the points (0:0:1:1) and (3:0:1:1). A parallel line passes through points (0:0:0:1) and (3:0:0:1). This second line intersects the plane at infinity at the point (3:0:0:0). But the first line also passes through this point:

when . ■

Any pair of parallel planes in affine 3-space will intersect each other in a projective line (a line at infinity) in the plane at infinity. Also, every plane in the affine 3-space intersects the plane at infinity in a unique line. [3] This line is determined by the direction—and only by the direction—of the plane.

Properties

Since the plane at infinity is a projective plane, it is homeomorphic to the surface of a "sphere modulo antipodes", i.e. a sphere in which antipodal points are equivalent: S2/{1,-1} where the quotient is understood as a quotient by a group action (see quotient space).

Notes

  1. ↑ Samuel 1988 , p. 11
  2. ↑ Meserve 1983 , p. 150
  3. ↑ Woods 1961 , p. 187

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