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.
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In geometry, any hyperplane H of a projective space P may be taken as a hyperplaneatinfinity. Then the set complement P ∖ H is called an affine space...
projective hyperplane is the infinite or ideal hyperplane, which is defined with the set of all points atinfinity. In projective space, a hyperplane does not...
that projective space that leave the hyperplaneatinfinity invariant, restricted to the complement of that hyperplane. A generalization of an affine transformation...
also refers to a line atinfinity in plane geometry, a plane atinfinity in three-dimensional space, and a hyperplaneatinfinity for general dimensions...
depending on the dimension of the space, the line atinfinity, the plane atinfinity or the hyperplaneatinfinity, in all cases a projective space of one less...
points atinfinity, which consists of homogenizing the defining polynomials, and removing the components that are contained in the hyperplaneatinfinity, by...
complement of a hyperplane. The projective completion is unique up to an isomorphism. The hyperplane is called the hyperplaneatinfinity, and its points...
arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S. Questions about a hyperplane arrangement...
one chooses the hyperplane of equation x 0 = 0 {\displaystyle x_{0}=0} as hyperplaneatinfinity, the complement of this hyperplane is an affine space...
respecting (mapping to itself, not fixing pointwise) the chosen hyperplaneatinfinity. This subgroup has a known structure (semidirect product of the...
affine plane (or affine space) plus a line (hyperplane) "atinfinity" and then treating that line (or hyperplane) as "ordinary". An algebraic model for doing...
typical for an affine variety to acquire singular points on the hyperplaneatinfinity, when its closure in projective space is taken. Resolution says...
In projective geometry, affine space means the complement of a hyperplaneatinfinity in a projective space. Affine space can also be viewed as a vector...
subspaces of the space obtained by treating this fixed hyperplane as the hyperplaneatinfinity. In the conventions common in applications to quantum groups...
proceed from Pn to the affine space An by declaring a hyperplane ω to be a hyperplaneatinfinity, we obtain the affine group A {\displaystyle {\mathfrak...
bundle (i.e., the structure sheaf). Then P(E) is a hyperplane in P(E ⊕ 1), called the hyperplaneatinfinity, and the complement of P(E) can be identified...
In that case, the intersection point mentioned above lies on the hyperplaneatinfinity. Affine spheres have been the subject of much investigation, with...
space that preserve affine space (equivalently, that leave the hyperplaneatinfinity invariant as a set) yield transformations of affine space. Conversely...
hyperplaneat a given point, as their radii go towards infinity. In Euclidean geometry, such a "hypersphere of infinite radius" would be a hyperplane...
complex projective space by fixing a hyperplane, which can be thought of as a hyperplane of ideal points "atinfinity" of the affine space. To illustrate...
space of n+1 dimensions, and regard the landscape to be painted as a hyperplane in this space. Suppose that the eye of the artist is the origin in Rn+1...
dimension n is the number of intersection points of the variety with n hyperplanes in general position. For an algebraic set, the intersection points must...
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