Hyperplane

[1] A reflection across a hyperplane is a kind of motion (geometric transformation preserving distance between points), and the group of all motions is generated by the reflections.

In non-Euclidean geometry, the ambient space might be the n-dimensional sphere or hyperbolic space, or more generally a pseudo-Riemannian space form, and the hyperplanes are the hypersurfaces consisting of all geodesics through a point which are perpendicular to a specific normal geodesic.

For example, in affine space, there is no concept of distance, so there are no reflections or motions.

In greatest generality, the notion of hyperplane is meaningful in any mathematical space in which the concept of the dimension of a subspace is defined.

The difference in dimension between a subspace and its ambient space is known as its codimension.

Several specific types of hyperplanes are defined with properties that are well suited for particular purposes.

In Cartesian coordinates, such a hyperplane can be described with a single linear equation of the following form (where at least one of the

Any hyperplane of a Euclidean space has exactly two unit normal vectors:

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

The reason for this is that the space essentially "wraps around" so that both sides of a lone hyperplane are connected to each other.

In convex geometry, two disjoint convex sets in n-dimensional Euclidean space are separated by a hyperplane, a result called the hyperplane separation theorem.

In machine learning, hyperplanes are a key tool to create support vector machines for such tasks as computer vision and natural language processing.

The datapoint and its predicted value via a linear model is a hyperplane.

In astronomy, hyperplanes can be used to calculate shortest distance between star systems, galaxies and celestial bodies with regard of general relativity and curvature of space-time as optimized geodesic or paths influenced by gravitational fields.

The theory of polyhedra and the dimension of the faces are analyzed by looking at these intersections involving hyperplanes.