Oregon Caves National Monument: Difference between revisions

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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 <math>(x_1, ..., x_n, x_{n+1})</math> are [[homogeneous coordinates]] for n-dimensional projective space, then the equation <math>x_{n+1} = 1</math> defines a hyperplane at infinity for the n-dimensional affine space with coordinates <math>(x_1, ..., x_n)</math>. H may also be called the '''ideal hyperplane'''.


Similarly, starting from an affine space '''A''', every class of [[parallel (geometry)|parallel]] lines can be associated with a [[point at infinity]]. The [[union (set theory)|union]] over all classes of parallels constitutes a 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 <math> \mathbb{R}P^n</math>.  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 space|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).
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In the projective space, each projective subspace of dimension ''k'' intersects the ideal hyperplane in a projective subspace "at infinity" whose dimension is ''k'' &minus; 1.
 
A pair of non-[[parallel (geometry)|parallel]] affine hyperplanes intersect at an affine subspace of dimension ''n'' &minus; 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.
 
==See also==
* [[Line at infinity]]
* [[Plane at infinity]]
* [[Affine sphere]]
 
==References==
* Albrecht Beutelspacher & Ute Rosenbaum (1998) ''Projective Geometry: From Foundations to Applications'', p 27, [[Cambridge University Press]] ISBN 0-521-48277-1 .
 
 
[[Category:Projective geometry]]
[[Category:Infinity]]

Latest revision as of 20:45, 19 February 2014


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