Hyperplane

An affine hyperplane is an affine subspace of codimension 1 in an affine space. In Cartesian coordinates, such a hyperplane can be described with a single linear equation of the following form (where at least one of the ${\displaystyle a_{i}}$s is non-zero and ${\displaystyle b}$ is an arbitrary constant):

${\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}=b.\ }$

In the case of a real affine space, in other words when the coordinates are real numbers, this affine space separates the space into two half-spaces, which are the connected components of the complement of the hyperplane, and are given by the inequalities

${\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}<b\ }$

and

${\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}>b.\ }$

As an example, a point is a hyperplane in 1-dimensional space, a line is a hyperplane in 2-dimensional space, and a plane is a hyperplane in 3-dimensional space. A line in 3-dimensional space is not a hyperplane, and does not separate the space into two parts (the complement of such a line is connected).

Any hyperplane of a Euclidean space has exactly two unit normal vectors: ${\displaystyle \pm {\hat {n}}}$. In particular, if we consider ${\displaystyle \mathbb {R} ^{n+1}}$ equipped with the conventional inner product (dot product), then one can define the affine subspace with normal vector ${\displaystyle {\hat {n}}}$ and origin translation ${\displaystyle {\tilde {b}}\in \mathbb {R} ^{n+1}}$ as the set of all ${\displaystyle x\in \mathbb {R} ^{n+1}}$ such that ${\displaystyle {\hat {n}}\cdot (x-{\tilde {b}})=0}$.

Affine hyperplanes are used to define decision boundaries in many machine learning algorithms such as linear-combination (oblique) decision trees, and perceptrons.

In a vector space, a vector hyperplane is a subspace of codimension 1, only possibly shifted from the origin by a vector, in which case it is referred to as a flat. Such a hyperplane is the solution of a single linear equation.

Projective hyperplanes, are used in projective geometry. A projective subspace is a set of points with the property that for any two points of the set, all the points on the line determined by the two points are contained in the set.[2] Projective geometry can be viewed as affine geometry with vanishing points (points at infinity) added. An affine hyperplane together with the associated points at infinity forms a projective hyperplane. One special case of a projective hyperplane is the infinite or ideal hyperplane, which is defined with the set of all points at infinity.

In projective space, a hyperplane does not divide the space into two parts; rather, it takes two hyperplanes to separate points and divide up the space. The reason for this is that the space essentially "wraps around" so that both sides of a lone hyperplane are connected to each other.

A hyperplane H is called a "support" hyperplane of the polyhedron P if P is contained in one of the two closed half-spaces bounded by H and ${\displaystyle H\cap P\neq \varnothing }$.[3] The intersection of P and H is defined to be a "face" of the polyhedron. The theory of polyhedra and the dimension of the faces are analyzed by looking at these intersections involving hyperplanes.