geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, any hyperplane ''H'' of a projective space ''P'' may be taken as a hyperplane at infinity. Then the

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

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 related ...

. 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 Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of ...

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 Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...

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 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 ...

''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-

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 is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pre ...

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

See also

References