Hyperplanes

Hyperplanes
Projection on a hyperplane
Geometry
Half-spaces

Hyperplanes

A hyperplane is a set described by a single scalar product equality. Precisely, an hyperplane in $mathbf{R}^n$ is a set of the form

$mathbf{H} = left{ x ~:~ a^Tx = b right},$

where $a in mathbf{R}^n$ , a ne 0 , and b in mathbf{R} are given. When b=0 , the hyperplane is simply the set of points that are orthogonal to ; when bne 0 , the hyperplane is a translation, along direction , of that set.

If $x_0 in mathbf{H}$ , then for any other element x in mathbf{H} , we have

Hence, the hyperplane can be characterized as the set of vectors such that x-x_0 is orthogonal to :

$mathbf{H} = left{ x ~:~ a^T(x-x_0)=0 right}.$

Hyperplanes are affine sets, of dimension n-1 (see the proof here). Thus, they generalize the usual notion of a plane in $mathbf{R}^3$ . Hyperplanes are very useful because they allows to separate the whole space in two regions. The notion of half-space formalizes this.

Example:

A hyperplane in $mathbf{R}^3$ .

Projection on a hyperplane

Consider the hyperplane $mathbf{H} = left{ x ~:~ a^Tx = b right}$ , and assume without loss of generality that is normalized ( |a|_2 = 1 ). We can represent as the set of points such that x-x_0 is orthogonal to , where x_0 is any vector in , that is, such that a^Tx_0 = b . One such vector is $x_{rm proj} := ba$ .

By construction, $x_{rm proj}$ is the projection of on mathbf{H} . That is, it is the point on closest to the origin, as it solves the projection problem

$min_x : |x|_2 ~:~ x in mathbf{H}.$

Indeed, for any x in mathbf{H} , using the Cauchy-Schwartz inequality:

|x_0|_2 = |b| = |a^Tx| le |a|_2 cdot |x|_2 = |x|_2,

and the minimum length |b| is attained with $x_{rm proj} = ba$ .

Geometry of hyperplanes

Geometrically, an hyperplane $mathbf{H} = left{ x ~:~ a^Tx = b right}$ , with |a|_2=1 , is a translation of the set of vectors orthogonal to . The direction of the translation is determined by , and the amount by .

Precisely, |b| is the length of the closest point x_0 on mathbf{H} from the origin, and the sign of determines if is away from the origin along the direction or . As we increase the magnitude of , the hyperplane is shifting further away along pm a , depending on the sign of . In the image on the left, the scalar is positive, as x_0 and point to the same direction.

Half-spaces

A half-space is a subset of $mathbf{R}^n$ defined by a single inequality involving a scalar product. Precisely, an half-space in $mathbf{R}^n$ is a set of the form

$mathbf{H} = left{ x ~:~ a^Tx ge b right},$

where $a in mathbf{R}^n$ , a ne 0 , and b in mathbf{R} are given.

Geometrically, the half-space above is the set of points such that a^T(x-x_0) ge 0 , that is, the angle between x-x_0 and is acute (in [-90^circ, +90^circ] ). Here x_0 is the point closest to the origin on the hyperplane defined by the equality a^Tx = b . (When is normalized, as in the picture, x_0 = ba .)

The half-space $left{ x ~:~ a^Tx ge b right}$ is the set of points such that x-x_0 forms an acute angle with , where x_0 is the projection of the origin on the boundary of the half-space.