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Subspace of n-space whose dimension is (n-1)
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like
Hyperplane
On the existence of hyperplanes separating disjoint convex sets
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar
Hyperplane_separation_theorem
Hyperplane in geometry
geometry, a supporting hyperplane of a set S {\displaystyle S} in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is a hyperplane that has both of the
Supporting_hyperplane
Concept in geometry
In geometry, any hyperplane H of a projective space P may be taken as a hyperplane at infinity. Then the set complement P ∖ H is called an affine space
Hyperplane_at_infinity
In mathematics, a hyperplane section of a subset X of projective space Pn is the intersection of X with some hyperplane H. In other words, we look at
Hyperplane_section
Partition of space by hyperplanes
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
Arrangement_of_hyperplanes
Set of methods for supervised statistical learning
hyperplane. This is called a linear classifier. There are many hyperplanes that might classify the data. One reasonable choice as the best hyperplane
Support_vector_machine
Bisection of Euclidean space by a hyperplane
two parts into which a hyperplane divides an n-dimensional space. That is, the points that are not incident to the hyperplane are partitioned into two
Half-space_(geometry)
Geometric property of a pair of sets of points in Euclidean geometry
is replaced by a hyperplane. The problem of determining if a pair of sets is linearly separable and finding a separating hyperplane if they are, arises
Linear_separability
Mapping from a Euclidean space to itself
a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension
Reflection_(mathematics)
Period of time occurring now
sometimes represented as a hyperplane in space-time, typically called "now", although modern physics demonstrates that such a hyperplane cannot be defined uniquely
Present
Vector bundle existing over a Grassmannian
dual of the hyperplane bundle or Serre's twisting sheaf O P n ( 1 ) {\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(1)} . The hyperplane bundle is the
Tautological_bundle
Algebraic structure in linear algebra
any subspace of dimension 1 less, i.e., of dimension n−1 is called a hyperplane. The counterpart to subspaces are quotient vector spaces. Given any subspace
Vector_space
Manifold or algebraic variety of dimension n in a space of dimension n+1
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety
Hypersurface
Set of statistical processes for estimating the relationships among variables
the unique line (or hyperplane) that minimizes the sum of squared differences between the true data and that line (or hyperplane). For specific mathematical
Regression_analysis
Topics referred to by the same term
valued Support (measure theory), a subset of a measurable space Supporting hyperplane, sometimes referred to as support Support of a module, a set of prime
Support
Theorem in algebraic geometry
specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape
Lefschetz_hyperplane_theorem
Geometric transformation combining reflection and translation
consists of a reflection across a hyperplane and a translation ("glide") in a direction parallel to that hyperplane, combined into a single transformation
Glide_reflection
Branch of geometry
is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete
Contact_geometry
Theorem on extension of bounded linear functionals
guise the theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem and has numerous uses in convex geometry. The theorem
Hahn–Banach_theorem
Concept in linear algebra
a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was used in a 1958
Householder_transformation
Algebraic geometry theorem
of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields
Theorem_of_Bertini
Path of an object through spacetime
related by dudτ=dwdτ, then they share the same simultaneous hyperplane. This hyperplane exists mathematically, but physical relations in relativity involve
World_line
Distance from a data point to a decision boundary
a given dataset, there may be many hyperplanes that could classify it. One reasonable choice as the best hyperplane is the one that represents the largest
Margin_(machine_learning)
Property of a mathematical space
that the intersection of a variety with a hyperplane reduces the dimension by one unless if the hyperplane contains the variety. An algebraic set being
Dimension
is the intersection of all hyperplanes that divide X {\displaystyle X} into two parts of equal moment about the hyperplane. Informally, it is the "average"
List_of_centroids
Fundamental space of geometry
space of dimension n is a set of n + 1 points that are not contained in a hyperplane. An affine basis defines barycentric coordinates for every point. Many
Euclidean_space
Machine learning strategy
n-dimensional distance from that datum to the separating hyperplane. Minimum Marginal Hyperplane methods assume that the data with the smallest W are those
Active learning (machine learning)
Active_learning_(machine_learning)
Generalized sphere of dimension n (mathematics)
stereographic projection, an n-sphere can be mapped onto an n-dimensional hyperplane by the n-dimensional version of the stereographic projection. For example
N-sphere
Hypersurface used by a classification algorithm
output label of a classifier is ambiguous. If the decision surface is a hyperplane, then the classification problem is linear, and the classes are linearly
Decision_boundary
Regular tessellation in 4D Euclidean space
some hyperplane orthogonal to a line joining opposite vertices of one of the 24-cells. For instance, one could take any of the coordinate hyperplanes in
24-cell_honeycomb
Distance from origin of tangent hyperplanes
{\displaystyle \mathbb {R} ^{n}} describes the (signed) distances of supporting hyperplanes of A from the origin. The support function is a convex function on R
Support_function
Concept in algebraic geometry
a hyperplane in P n {\displaystyle \mathbb {P} ^{n}} (because the zero set of a section of O ( 1 ) {\displaystyle {\mathcal {O}}(1)} is a hyperplane).
Ample_line_bundle
N-dimensional generalisation of a pyramid
(n – 1)-polytope in a (n – 1)-dimensional hyperplane. A point called apex is located outside the hyperplane and gets connected to all the vertices of
Hyperpyramid
Equation that does not involve powers or products of variables
More generally, the solutions of a linear equation in n variables form a hyperplane (a subspace of dimension n − 1) in the Euclidean space of dimension n
Linear_equation
Minkowksi sum of line segments
as a projection of a hypercube. Zonotopes are intimately connected to hyperplane arrangements and matroid theory. The Minkowski sum of a finite set of
Zonotope
Abstraction of ordered linear algebra
properties of directed graphs, vector arrangements over ordered fields, and hyperplane arrangements over ordered fields. In comparison, an ordinary (i.e., non-oriented)
Oriented_matroid
Theorem that any three objects in space can be simultaneously bisected by a plane
respect to their measure, e.g. volume) with a single (n − 1)-dimensional hyperplane. This is possible even if the objects overlap. It was proposed by Hugo
Ham_sandwich_theorem
Sphere-like surface
{\mathcal {O}}} in at most 2 points, The tangents at a point cover a hyperplane (and nothing more), and O {\displaystyle {\mathcal {O}}} contains no lines
Ovoid_(projective_geometry)
Concept in mathematics
generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise. Complex reflection groups arise in the study of the invariant
Complex_reflection_group
Invariant measure of fractal dimension
Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope
Hausdorff_dimension
Geometric transformation that preserves lines but not angles nor the origin
that projective space that leave the hyperplane at infinity invariant, restricted to the complement of that hyperplane. A generalization of an affine transformation
Affine_transformation
Hypersurface in hyperbolic space
limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of tangency. For n = 2 a horosphere is called a horocycle
Horosphere
Type of convex polytope
with cut hyperplanes passing through these coordinates. A d-polytope requires at least d + 1 vertices, and can't be all in the same hyperplanes. n-simplex
0/1-polytope
Multivariate generalization of the median
d-dimensional space, a centerpoint of the set is a point such that any hyperplane that goes through that point divides the set of points in two roughly
Centerpoint_(geometry)
In mathematics, dimension of a ring
Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope
Krull_dimension
Economic Model
Define the price hyperplane ⟨p,q⟩=⟨p,∑jyj⟩. Since it's a supporting hyperplane of 𝑃𝑃𝑆, and 𝑃𝑃𝑆 is a convex cone, the price hyperplane passes the origin
Arrow–Debreu_model
-dimensional unit hypercube [ 0 , 1 ] d {\displaystyle [0,1]^{d}} with the hyperplane of equation x 1 + ⋯ + x d = k {\displaystyle x_{1}+\cdots +x_{d}=k} and
Hypersimplex
Mathematical concept
plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity
Infinity
In algebraic geometry, a relative effective Cartier divisor is roughly a family of effective Cartier divisors. Precisely, an effective Cartier divisor
Relative effective Cartier divisor
Relative_effective_Cartier_divisor
Multidimensional search tree for points in k dimensional space
generating a splitting hyperplane that divides the space into two parts, known as half-spaces. Points to the left of this hyperplane are represented by the
K-d_tree
Algorithmic technique using hashing
hyperplane (defined by a normal unit vector r) at the outset and use the hyperplane to hash input vectors. Given an input vector v and a hyperplane defined
Locality-sensitive_hashing
Mathematical concept
space of n+1 dimensions, and regard the landscape to be painted as a hyperplane in this space. Suppose that the eye of the artist is the origin in Rn+1
Complex_projective_space
Subgroup of a root system's isometry group
Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to at least one of the roots, and as such is a finite reflection
Weyl_group
Invariant of topological spaces
Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope
Inductive_dimension
Method for recursively subdividing a space into two subsets using hyperplanes
recursively subdivides a Euclidean space into two convex sets by using hyperplanes as partitions. This process of subdividing gives rise to a representation
Binary_space_partitioning
Length in solid geometry
consequence of the Cauchy–Schwarz inequality. The vector equation for a hyperplane in n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle
Distance from a point to a plane
Distance_from_a_point_to_a_plane
Conic solid with a polygonal base
− 1)-polytope in a (n − 1)-dimensional hyperplane. A point called the apex is located outside the hyperplane and gets connected to all the vertices of
Pyramid_(geometry)
Solvability theorem for finite systems of linear inequalities
than 90°. The hyperplane normal to this vector has the vectors ai on one side and the vector b on the other side. Hence, this hyperplane separates the
Farkas'_lemma
Arrangement of hyperplanes
In mathematics, a supersolvable arrangement is a hyperplane arrangement that has a maximal flag consisting of modular elements. Equivalently, the intersection
Supersolvable_arrangement
Geometric model of the physical space
parallel to the given line. A hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space
Three-dimensional_space
Mathematical space with two coordinates
Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope
Two-dimensional_space
Sum of terms, each multiplied with a scalar
non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes
Linear_combination
Mathematical object
intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere (unless the hyperplane is tangent to the 3-sphere, in which case the intersection
3-sphere
Concept in linear algebra
a⊥b), is the orthogonal projection of a onto the plane (or, in general, hyperplane) that is orthogonal to b. Since both proj b a {\displaystyle \operatorname
Vector_projection
Russian-born American mathematician (1884–1972)
Paris at the École Centrale Paris. He proved theorems on the topology of hyperplane sections of algebraic varieties, which provide a basic inductive tool
Solomon_Lefschetz
Machine learning algorithm
Euclidean, though others may be used) of a sample from the separating hyperplane is the margin of that sample. The notion of margins is important in several
Margin_classifier
Topics referred to by the same term
space of dimension 3n/2 + 2 that can be isomorphically projected to a hyperplane. This disambiguation page lists mathematics articles associated with the
Severi_variety
Motion of a certain space that preserves at least one point
other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of
Rotation_(mathematics)
Knot theory
(2k+1)-dimensional contact manifold that is always tangent to the contact hyperplane. Two smooth knots are equivalent if there is a way to smoothly deform
Legendrian_knot
Hungarian and American mathematician and physicist (1903–1957)
represent prices and quantities, the use of supporting and separating hyperplanes and convex sets, and fixed-point theory—have been primary tools of mathematical
John_von_Neumann
Line or vector perpendicular to a curve or a surface
point on the hyperplane and 𝐯i for i=1,…,n−1 are linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector
Normal_(geometry)
Generalizations of codimension-1 subvarieties of algebraic varieties
An over k is equal to zero. Since projective space Pn over k minus a hyperplane H is isomorphic to An, it follows that the divisor class group of Pn is
Divisor_(algebraic_geometry)
Topics referred to by the same term
a cube Hyperoperation, an arithmetic operation beyond exponentiation Hyperplane, a subspace whose dimension is one less than that of its ambient space
Hyper
Faster-than-light travel in science fiction
of Element 117 (1949) by Milton Smith, a window is opened into a new "hyperplane of hyperspace" containing those who have already died on Earth, and similarly
Hyperspace
optimization, the supporting functional is a generalization of the supporting hyperplane of a set. Let X be a locally convex topological space, and C ⊂ X {\displaystyle
Supporting_functional
Points of small height in projective space lie in a finite number of hyperplanes
points of small height in projective space lie in a finite number of hyperplanes. It is a result obtained by Wolfgang M. Schmidt (1972). The subspace
Subspace_theorem
Mathematical description of spacetime used in relativity
which is a hyperplane passing through the origin, and its norm. Geometrically thus, covariant vectors should be viewed as a set of hyperplanes, with spacing
Minkowski_spacetime
Family of polynomials
fixing a hyperplane, counting such subspaces contained in that hyperplane, and then counting the subspaces not contained in the hyperplane; these latter
Gaussian_binomial_coefficient
Discrete group type in group theory
group of E which is generated by a set of orthogonal reflections across hyperplanes passing through the origin. An affine reflection group is a discrete
Reflection_group
Number of vectors in any basis of the vector space
Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope
Dimension_(vector_space)
Concept in projective geometry
plane at infinity is the hyperplane at infinity of a three dimensional projective space or to any plane contained in the hyperplane at infinity of any projective
Plane_at_infinity
Geometric space with six dimensions
Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope
Six-dimensional_space
Geometric space with five dimensions
Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope
Five-dimensional_space
In geometry, set whose intersection with every line is a single line segment
half-spaces (sets of points in space that lie on and to one side of a hyperplane). From what has just been said, it is clear that such intersections are
Convex_set
Linear map from a vector space to its field of scalars
of mutually parallel planes; in higher dimensions, they are parallel hyperplanes. This method of visualizing linear functionals is sometimes introduced
Linear_form
Branch of mathematics
open problem in convex geometry and asymptotic geometric analysis is the hyperplane conjecture, also known as the slicing problem. It asks whether there exists
Asymptotic_geometry
Mathematics of convex functions and sets
minimum is also a global minimum. Convex sets can often be separated by hyperplanes, and convex functions can be studied through supporting affine functions
Convex_analysis
1960–67 foundational treatise on algebraic geometry by Alexander Grothendieck
essentially complete; some changes made in last sections; the section on hyperplane sections made into the new Chapter V of second edition (draft exists)
Éléments de géométrie algébrique
Éléments_de_géométrie_algébrique
Overview of and topical guide to geometry
Convex Convex hull Coxeter group Euclidean distance Homothetic center Hyperplane Lattice Ehrhart polynomial Leech lattice Minkowski's theorem Packing Sphere
Outline_of_geometry
Type of geometric transformation
{\displaystyle \mathbb {R} ^{n},} the distance is measured from a fixed hyperplane parallel to the direction of displacement. This geometric transformation
Shear_mapping
Theorem in probability theory
probabilistic restatement of Schläfli's theorem that N {\displaystyle N} hyperplanes in general position in R n {\displaystyle \mathbb {R} ^{n}} divides it
Wendel's_theorem
Dimensionality of space at which the character of the phase transition changes
{\displaystyle E_{i,j}} lie on a hyperplane, for examples see the figure above. N {\displaystyle N} is a normal vector of this hyperplane. The lower critical dimension
Critical_dimension
Statistical method
example, the hyperplane is just a 2-dimensional plane defined by the two factor vectors. The projection of the data vectors onto the hyperplane is given by
Factor_analysis
Property of a space in which the local dimensionality is the same everywhere
Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope
Equidimensionality
All bundles a consumer can afford based on various conditions
budget set is bounded above by a k {\displaystyle k} -dimensional budget hyperplane characterized by the equation p x = m {\displaystyle \mathbf {p} \mathbf
Budget_set
Thing in mathematics and theoretical physics
physics, a quasi-sphere is a generalization of the hypersphere and the hyperplane to the context of a pseudo-Euclidean space. It may be described as the
Quasi-sphere
Integral transform in mathematics
{\displaystyle Rf} on the space Σ n {\displaystyle \Sigma _{n}} of all hyperplanes in R n {\displaystyle \mathbb {R} ^{n}} . It is defined by: R f ( ξ )
Radon_transform
Construction in group theory
Rotations in axes perpendicular to the hyperplane preserve the hyperplane and yield a rotation of the hyperplane (an element of SO(n), which has dimension
Projective_linear_group
Uniform 4-polytope bounded by 320 cells
with transparent triangular faces Orthographic projection Centered on hyperplane of an antiprism in one of the two rings. 3D orthographic projection of
Grand_antiprism
travel, tourism, insurance
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travel, tourism, insurance