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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
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
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
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
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
Partition of space by a 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
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
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
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)
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
Algebraic structure in linear algebra
dimension 1 less, i.e., of dimension n − 1 {\displaystyle n-1} is called a hyperplane. The counterpart to subspaces are quotient vector spaces. Given any subspace
Vector_space
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
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
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
Path of an object through spacetime
}}={\frac {dw}{d\tau }},} then they share the same simultaneous hyperplane. This hyperplane exists mathematically, but physical relations in relativity involve
World_line
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
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)
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
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
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)
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
{\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 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
-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
Theorem on extension of bounded linear functionals
Hahn–Banach 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
Minkowsi 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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
Four-dimensional number system
Inductive Hausdorff Minkowski Fractal Degrees of freedom Polytopes and shapes Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope
Quaternion
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
Geometric object with flat sides
Regular polytopes, p. 127 The part of the polytope that lies in one of the hyperplanes is called a cell Beck, Matthias; Robins, Sinai (2007), Computing the
Polytope
Significant topic in economics
points in Q. Supporting hyperplane is a concept in geometry. A hyperplane divides a space into two half-spaces. A hyperplane is said to support a set
Convexity_in_economics
Economic Model
Proof sketch The price hyperplane separates the attainable productions and the Pareto-better consumptions. That is, the hyperplane ⟨ p ∗ , q ⟩ = ⟨ p ∗
Arrow–Debreu_model
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
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
Abstraction of linear independence of vectors
r-1} is called a hyperplane, or co-atoms or copoints. These are the maximal proper flats; that is, the only superset of a hyperplane that is also a flat
Matroid
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
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
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
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
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
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)
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
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
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
Completion of the usual space with "points at infinity"
any n + 1 of them are independent; that is, they are not contained in a hyperplane. If V is an (n + 1)-dimensional vector space, and p is the canonical projection
Projective_space
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
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
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
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
Loss function in machine learning
( w , b ) {\displaystyle (\mathbf {w} ,b)} are the parameters of the hyperplane and x {\displaystyle \mathbf {x} } is the input variable(s). When t and
Hinge_loss
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
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
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
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)
Convex hull of a finite set of points in a Euclidean space
with a supporting hyperplane of the polytope, a hyperplane bounding a half-space that contains the polytope. If a supporting hyperplane also intersects
Convex_polytope
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
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
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
Line or vector perpendicular to a curve or a surface
n-1} are linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector n {\displaystyle \mathbf {n} } in the null
Normal_(geometry)
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
Generalized sphere of dimension n (mathematics)
n} -sphere can be mapped onto an n {\displaystyle n} -dimensional hyperplane by the n {\displaystyle n} -dimensional version of the stereographic
N-sphere
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
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
Decomposition into connected open cells of lower dimensions, by a finite set of objects
dimension of the space, and often of the same type as each other, such as hyperplanes or spheres. For a set A {\displaystyle A} of objects in R d {\displaystyle
Arrangement_(space_partition)
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
Theosophical philosophical concept
It represents the fourth [higher] subplane of the physical plane (a hyperplane), the lower three being the states of solid, liquid, and gaseous matter
Etheric_plane
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
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
Geometric arrangements of points, foundational to Lie theory
the set Φ {\displaystyle \Phi } is closed under reflection through the hyperplane perpendicular to α {\displaystyle \alpha } . (Integrality) If α {\displaystyle
Root_system
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
Group that admits a formal description in terms of reflections
given two hyperplanes meeting at an angle of π / k {\displaystyle \pi /k} , the composite of the two reflections about these hyperplanes is a rotation
Coxeter_group
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
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
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
Type of vector space in math
closed convex set can be separated from any point outside it by means of a hyperplane of the Hilbert space. This is an immediate consequence of the best approximation
Hilbert_space
Equality of areas of a sliced disk
dimensions, i.e. for certain arrangements of hyperplanes, the alternating sum of volumes cut out by the hyperplanes is zero. Compare with the ham sandwich theorem
Pizza_theorem
Coordinate system that is defined by points instead of vectors
the complement of a hyperplane. The projective completion is unique up to an isomorphism. The hyperplane is called the hyperplane at infinity, and its
Barycentric_coordinate_system
Problem in convex geometry
asks whether it is true that a symmetric convex body with larger central hyperplane sections has larger volume. More precisely, if K, T are symmetric convex
Busemann–Petty_problem
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
Division of an entire space into ≥2 disjoint subsets
Most space-partitioning systems use planes (or, in higher dimensions, hyperplanes) to divide space: points on one side of the plane form one region, and
Space_partitioning
Matroid with graph forests as independent sets
be realized as the lattice of a hyperplane arrangement, in fact as a subset of the braid arrangement, whose hyperplanes are the diagonals H i j = { ( x
Graphic_matroid
n-dimensional Euclidean space such that whenever K and L are projected onto a hyperplane, the volume of the projection of K is smaller than the volume of the projection
Shephard's_problem
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
Polyhedron whose vertices represent permutations
The permutohedron of order n lies entirely in the (n − 1)-dimensional hyperplane consisting of all points whose coordinates sum to the number: 1 + 2 +
Permutohedron
HYPERPLANE
HYPERPLANE
HYPERPLANE
HYPERPLANE
Male
Ukrainian
, man, warrior.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Marathi, Tamil, Telugu
Lord Venkateshwara
Girl/Female
Basque Spanish
Angel.
Girl/Female
Tamil
Karishma | கரிஷà¯à®®à®¾
Favor: gift, Miracle
Girl/Female
Hindu
Required victory, Superior
Girl/Female
Indian
Boy/Male
Tamil
Dousik | தோஉஂஸிக
Intelligent
Boy/Male
Indian, Sanskrit, Tamil
Young Ascetic
Boy/Male
African, American, Australian, British, Chinese, Christian, English, Indian, Jamaican
Settlement in a Grassy Place; Bent Grass Enclosure; Moor Dweller; Bent Grass Settlement
Boy/Male
British, English
From the Valley
HYPERPLANE
HYPERPLANE
HYPERPLANE
HYPERPLANE
HYPERPLANE