Ingeometry, ahyperplane is a generalization of atwo-dimensional plane inthree-dimensional space tomathematical spaces of arbitrarydimension. Like aplane in space, a hyperplane is aflathypersurface, asubspace whosedimension is one less than that of theambient space. Two lower-dimensional examples of hyperplanes areone-dimensionallines in a plane andzero-dimensionalpoints on a line.
Most commonly, the ambient space isn-dimensionalEuclidean space, in which case the hyperplanes are the(n − 1)-dimensional"flats", each of which separates the space into twohalf spaces.[1] Areflection across a hyperplane is a kind ofmotion (geometric transformation preservingdistance between points), and thegroup of all motions isgenerated by the reflections. Aconvex polytope is theintersection of half-spaces.
Innon-Euclidean geometry, the ambient space might be then-dimensional sphere orhyperbolic space, or more generally apseudo-Riemannianspace form, and the hyperplanes are the hypersurfaces consisting of allgeodesics through a point which areperpendicular to a specificnormal geodesic.
In other kinds of ambient spaces, some properties from Euclidean space are no longer relevant. For example, inaffine space, there is no concept of distance, so there are no reflections or motions. In anon-orientable space such aselliptic space orprojective space, there is no concept of half-planes. In greatest generality, the notion of hyperplane is meaningful in any mathematical space in which the concept of the dimension of asubspace is defined.
The difference in dimension between a subspace and its ambient space is known as itscodimension. A hyperplane has codimension1.
Ingeometry, ahyperplane of ann-dimensional spaceV is a subspace of dimensionn − 1, or equivalently, ofcodimension 1 in V. The spaceV may be aEuclidean space or more generally anaffine space, or avector space or aprojective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can be given incoordinates as the solution of a single (due to the "codimension 1" constraint)algebraic equation of degree 1.
IfV is a vector space, one distinguishes "vector hyperplanes" (which arelinear subspaces, and therefore must pass through the origin) and "affine hyperplanes" (which need not pass through the origin; they can be obtained bytranslation of a vector hyperplane). A hyperplane in a Euclidean space separates that space into twohalf spaces, and defines areflection that fixes the hyperplane and interchanges those two half spaces.
Several specific types of hyperplanes are defined with properties that are well suited for particular purposes. Some of these specializations are described here.
Anaffine hyperplane is anaffine subspace ofcodimension 1 in anaffine space.InCartesian coordinates, such a hyperplane can be described with a singlelinear equation of the following form (where at least one of thes is non-zero and is an arbitrary constant):
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 theconnected components of thecomplement of the hyperplane, and are given by theinequalities
and
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:. In particular, if we consider equipped with the conventional inner product (dot product), then one can define the affine subspace with normal vector and origin translation as the set of all such that.
Affine hyperplanes are used to define decision boundaries in manymachine learning algorithms such as linear-combination (oblique)decision trees, andperceptrons.
In a vector space, a vector hyperplane is asubspace of codimension 1, only possibly shifted from the origin by a vector, in which case it is referred to as aflat. Such a hyperplane is the solution of a singlelinear equation.
Projective hyperplanes are used inprojective geometry. Aprojective 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 asaffine geometry withvanishing 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 theinfinite orideal 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.
Inconvex geometry, twodisjointconvex sets in n-dimensional Euclidean space are separated by a hyperplane, a result called thehyperplane separation theorem.
Inmachine learning, hyperplanes are a key tool to createsupport vector machines for such tasks ascomputer vision andnatural language processing.
In multiplelinear regression with more than two regressors, the datapoint and its predicted value via a linear model is a hyperplane.
Inastronomy, hyperplanes can be used to calculate shortest distance between star systems, galaxies and celestial bodies with regard ofgeneral relativity and curvature ofspace-time as optimizedgeodesic or paths influenced by gravitational fields.
Thedihedral angle between two non-parallel hyperplanes of a Euclidean space is the angle between the correspondingnormal vectors. The product of the transformations in the two hyperplanes is arotation whose axis is thesubspace of codimension 2 obtained by intersecting the hyperplanes, and whose angle is twice the angle between the hyperplanes.
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.[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.
