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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