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).
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.