# Hyperplane at infinity

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 are homogeneous coordinates for n-dimensional projective space, then the equation defines a hyperplane at infinity for the n-dimensional affine space with coordinates . H may also be 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 constitutes a 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 . There is one ideal point added for each pair of opposite directions in **A**.

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.

## See also

## References

- Albrecht Beutelspacher & Ute Rosenbaum (1998)
*Projective Geometry: From Foundations to Applications*, p 27, Cambridge University Press ISBN 0-521-48277-1 .