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# Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. 

## Affine geometry

In mathematics, affine geometry is what remains of Euclidean geometry when not using (mathematicians often say "when forgetting") the metric notions of distance and angle.

## Affine space

In mathematics, an affine space is a geometric structure that generalizes the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

## Algebraic equation

In mathematics, an algebraic equation or polynomial equation is an equation of the form where P and Q are polynomials with coefficients in some field, often the field of the rational numbers.

## Allyn & Bacon

Allyn & Bacon, founded in 1868, is a higher education textbook publisher in the areas of education, humanities and social sciences.

## Ambient space

An ambient space or ambient configuration space is the space surrounding an object.

## American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

## Arrangement of hyperplanes

In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S. Questions about a hyperplane arrangement A generally concern geometrical, topological, or other properties of the complement, M(A), which is the set that remains when the hyperplanes are removed from the whole space.

## Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

## Charles W. Curtis

Charles Whittlesey Curtis (born October 13, 1926) is a mathematician and historian of mathematics, known for his work in finite group theory and representation theory.

## Codimension

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.

## Complement (set theory)

In set theory, the complement of a set refers to elements not in.

## Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

## Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

## Decision boundary

In a statistical-classification problem with two classes, a decision boundary or decision surface is a hypersurface that partitions the underlying vector space into two sets, one for each class.

## Decision tree learning

Decision tree learning uses a decision tree (as a predictive model) to go from observations about an item (represented in the branches) to conclusions about the item's target value (represented in the leaves).

## Dihedral angle

A dihedral angle is the angle between two intersecting planes.

## Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Flat (geometry)

In geometry, a flat is a subset of n-dimensional space that is congruent to a Euclidean space of lower dimension.

## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

## Half-space (geometry)

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space.

## Ham sandwich theorem

In mathematical measure theory, for every positive integer the ham sandwich theorem states that given measurable "objects" in -dimensional Euclidean space, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single -dimensional hyperplane.

## Heinrich Guggenheimer

Heinrich Walter Guggenheimer (born 21 July 1924) is a German-born American mathematician who has contributed to knowledge in differential geometry, topology, algebraic geometry, and convexity.

## Hyperplane separation theorem

In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space.

## Hypersurface

In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface.

## Inequality (mathematics)

In mathematics, an inequality is a relation that holds between two values when they are different (see also: equality).

## Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

## Linear equation

In mathematics, a linear equation is an equation that may be put in the form where x_1, \ldots, x_n are the variables or unknowns, and c, a_1, \ldots, a_n are coefficients, which are often real numbers, but may be parameters, or even any expression that does not contain the unknowns.

## Linear subspace

In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space.

## Machine learning

Machine learning is a subset of artificial intelligence in the field of computer science that often uses statistical techniques to give computers the ability to "learn" (i.e., progressively improve performance on a specific task) with data, without being explicitly programmed.

## Normal (geometry)

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.

## Perceptron

In machine learning, the perceptron is an algorithm for supervised learning of binary classifiers (functions that can decide whether an input, represented by a vector of numbers, belongs to some specific class or not).

## Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

## Projective geometry

Projective geometry is a topic in mathematics.

## Projective space

In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.

## Reflection (mathematics)

In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.

## Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry.

## Space (mathematics)

In mathematics, a space is a set (sometimes called a universe) with some added structure.

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## Supporting hyperplane

In geometry, a supporting hyperplane of a set S in Euclidean space \mathbb R^n is a hyperplane that has both of the following two properties.

## Translation (geometry)

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.

## Vanishing point

A vanishing point is a point on the image plane of a perspective drawing where the two-dimensional perspective projections (or drawings) of mutually parallel lines in three-dimensional space appear to converge.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## References

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