Lecture Notes

Hyperplanes

By Akshay Agrawal. Last updated February 3, 2019.

Hyperplane

A hyperplane is a set

where is nonzero and . The vector is referred to as the normal vector of the hyperplane, since, if is any point such that , the hyperplane can be expressed as

Euclidean projection

The Euclidean projection of a point onto the hyperplane is

One way to derive this solution is to analytically solve the optimization problem

using, for example, the method of Lagrange multiplers.

Distance between parallel hyperplanes

Consider two parallel hyperplanes and . The distance between the hyperplanes can be computed by projecting any point in the former hyperplane onto the latter hyperplane. In particular, the projection of onto is

Hence the distance between the two hyperplanes is

As an example, the width of the slab

is the distance between the hyperplanes and , which equals .