Hyperplane at infinity
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In mathematics, in particular projective geometry, the hyperplane at infinity, also called ideal hyperplane, is a projective (n − 1) -space added to Euclidean n-space — <math> \mathbb{R}^n <math> — in order to give it closure of incidence properties, thereby converting <math> \mathbb{R}^n <math> into the projective n-space <math> \mathbb{R}P^n<math>.
The completed projective space <math>\mathbb{R}P^n<math> contains projective hyperplanes of dimension n − 1, and each of these hyperplanes intersects the ideal hyperplane at a projective "hyperline at infinity" of dimension <math> n~-~2 <math>.
Moreover, a pair of non-parallel affine hyperplanes intersect at an affine hyperline, but a pair of parallel hyperplanes intersect at a projective hyperline which is a subspace — "lies on" — the projective hyperplane at infinity. Parallel hyperplanes intersect due to the addition of the hyperplane at infinity, and that is what is meant by "giving closure to incidence properties."
See also: point at infinity, line at infinity, plane at infinity.