About: Hyperplane at infinity

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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. 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.

Attributes	Values
rdfs:label	Hyperplane at infinity (en)
rdfs:comment	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. 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. (en)
dct:subject	Projective geometry Infinity
Wikipage page ID	713946 (xsd:integer)
Wikipage revision ID	1113900213 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Projective space Geometry Parallel (geometry) Line at infinity Affine space Projective geometry Hyperplane Infinity Homogeneous coordinates Union (set theory) Plane at infinity Point at infinity Affine subspace Set complement
sameAs	Hyperplane at infinity Hyperplane at infinity Hyperplane at infinity Hyperplane at infinity
dbp:wikiPageUsesTemplate	dbt:ISBN dbt:Short_description
has abstract	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. Similarly, starting from an affine space A, every class of parallel lines can be associated with a point at infinity. The union over all classes of parallels constitute the points of the hyperplane at infinity. Adjoining the points of this hyperplane (called ideal points) to A converts it into an n-dimensional projective space, such as the real projective space RPn. 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. (en)
prov:wasDerivedFrom	wikipedia-en:Hyperplane_at_infinity?oldid=1113900213&ns=0
page length (characters) of wiki page	2731 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Hyperplane_at_infinity
is Link from a Wikipage to another Wikipage of	List of algebraic geometry topics Barycentric coordinate system List of geometry topics Erlangen program Singularity theory Line at infinity Affine group Affine space Affine transformation Hypersurface Affine geometry Affine sphere Infinity Plane at infinity Point at infinity
is foaf:primaryTopic of	wikipedia-en:Hyperplane_at_infinity

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