geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, any

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

''H'' of a

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

''P'' may be taken as a hyperplane at infinity. Then the

set complement In set theory, the complement of a set , often denoted by A^c (or ), is the set of elements not in . When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set , the absolute complemen ...

is called an

affine 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 relat ...

. For instance, if are

homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...

for ''n''-dimensional projective space, then the equation defines a hyperplane at infinity for the ''n''-dimensional affine space with coordinates . ''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 In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...

. 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 . 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 In mathematics, an affine space is a geometry, geometric structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance (mathematics), distance ...

''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 . A pair of non- parallel affine hyperplanes intersect at an affine subspace of dimension , 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.

References

* Albrecht Beutelspacher & Ute Rosenbaum (1998) ''Projective Geometry: From Foundations to Applications'', p 27,

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

{{ISBN, 0-521-48277-1 . Projective geometry Infinity

See also

References