![]() ![]() A hyperplane in a Euclidean space separates that space into two half spaces, and defines a reflection that fixes the hyperplane and interchanges those two half spaces. If V is a vector space, one distinguishes "vector hyperplanes" (which are linear subspaces, and therefore must pass through the origin) and "affine hyperplanes" (which need not pass through the origin they can be obtained by translation of a vector hyperplane). The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings in all cases however, any hyperplane can be given in coordinates as the solution of a single (due to the "codimension 1" constraint) algebraic equation of degree 1. In geometry, a hyperplane of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V. Therefore, a necessary and sufficient condition for S to be a hyperplane in X is for S to have codimension one in X. The difference in dimension between a subspace S and its ambient space X is known as the codimension of S with respect to X. While a hyperplane of an n-dimensional projective space does not have this property. For instance, a hyperplane of an n-dimensional affine space is a flat subset with dimension n − 1 and it separates the space into two half spaces. In different settings, hyperplanes may have different properties. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. A plane is a hyperplane of dimension 2, when embedded in a space of dimension 3. Two intersecting planes in three-dimensional space. Albrecht Beutelspacher & Ute Rosenbaum (1998) Projective Geometry: From Foundations to Applications, p 27, Cambridge University Press ISBN 7-1.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. 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.Ī 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). 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). 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. 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 RP n.īy 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. The union over all classes of parallels constitute the points of the hyperplane at infinity. Similarly, starting from an affine space A, every class of parallel lines can be associated with a point at infinity. , x n, x n+1) are homogeneous coordinates for n-dimensional projective space, then the equation x n+1 = 0 defines a hyperplane at infinity for the n-dimensional affine space with coordinates ( x 1. Then the set complement P ∖ H is called an affine space. In geometry, any hyperplane H of a projective space P may be taken as a hyperplane at infinity. ![]()
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