Hyperplane at infinity
Encyclopedia
In mathematics
, in particular projective geometry
, the hyperplane at infinity, also called the ideal hyperplane, is an (n−1)-dimensional projective space added to an n-dimensional affine space
A, such as the real affine n-space
, in order to obtain uniformity of incidence
properties. Adding the points of this hyperplane (called ideal points or points at infinity) converts the affine space into an n-dimensional projective space
, such as the real projective space
. There is one ideal point added for each pair of opposite directions in A.
By adding these ideal points, the entire affine space A is completed to a projective space P, which may be called the projective completion of A. Each affine subspace S of A is completed to a projective subspace
of P by adding to S all the ideal points corresponding to the directions of the lines contained in S. The resulting projective subspaces are often called affine subspaces of the projective space P, as opposed to the infinite or ideal subspaces, which are the subspaces of the hyperplane at infinity (however, they are projective spaces, not affine spaces).
In the projective space, each projective subspace of dimension k intersects the ideal hyperplane in a projective subspace "at infinity" whose dimension is k − 1.
A pair of non-parallel
affine hyperplanes intersect at an affine subspace of dimension n − 2, but a parallel pair of affine hyperplanes intersect at a projective subspace of the ideal hyperplane (the intersection lies on the ideal hyperplane). Thus, parallel hyperplanes, which did not meet in the affine space, intersect in the projective completion due to the addition of the hyperplane at infinity.
Similarly, parallel lines intersect at the point at infinity which corresponds to their common direction.
In a projective space, any hyperplane may be chosen to be the hyperplane at infinity. Specifically, if P is a projective space and H is a hyperplane of P, then P − H is an affine space whose projective completion is P. Thus, the ideal hyperplane cannot be identified in terms of P alone.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in particular projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under 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 hyperplane at infinity, also called the ideal hyperplane, is an (n−1)-dimensional projective space added to an n-dimensional affine space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
A, such as the real affine n-space
, in order to obtain uniformity of incidenceIncidence
Incidence may refer to:* Incidence , a measure of the risk of developing some new condition within a specified period of time* Incidence , the binary relations describing how subsets meet...
properties. Adding the points of this hyperplane (called ideal points or points at infinity) converts the affine space into an n-dimensional projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
, such as the real projective space
. There is one ideal point added for each pair of opposite directions in A.By adding these ideal points, the entire affine space A is completed to a projective space P, which may be called the projective completion of A. Each affine subspace S of A is completed to a projective subspace
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
of P by adding to S all the ideal points corresponding to the directions of the lines contained in S. The resulting projective subspaces are often called affine subspaces of the projective space P, as opposed to the infinite or ideal subspaces, which are the subspaces of the hyperplane at infinity (however, they are projective spaces, not affine spaces).
In the projective space, each projective subspace of dimension k intersects the ideal hyperplane in a projective subspace "at infinity" whose dimension is k − 1.
A pair of non-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...
affine hyperplanes intersect at an affine subspace of dimension n − 2, but a parallel pair of affine hyperplanes intersect at a projective subspace of the ideal hyperplane (the intersection lies on the ideal hyperplane). Thus, parallel hyperplanes, which did not meet in the affine space, intersect in the projective completion due to the addition of the hyperplane at infinity.
Similarly, parallel lines intersect at the point at infinity which corresponds to their common direction.
In a projective space, any hyperplane may be chosen to be the hyperplane at infinity. Specifically, if P is a projective space and H is a hyperplane of P, then P − H is an affine space whose projective completion is P. Thus, the ideal hyperplane cannot be identified in terms of P alone.