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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
Subspace of n-space whose dimension is (n-1)
projective hyperplane is the infinite or ideal hyperplane, which is defined with the set of all points at infinity. In projective space, a hyperplane does not
Hyperplane
Mathematical concept
also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions
Infinity
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
Concept in geometry
depending on the dimension of the space, the line at infinity, the plane at infinity or the hyperplane at infinity, in all cases a projective space of one less
Point_at_infinity
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
Plane_at_infinity
Manifold or algebraic variety of dimension n in a space of dimension n+1
one chooses the hyperplane of equation x 0 = 0 {\displaystyle x_{0}=0} as hyperplane at infinity, the complement of this hyperplane is an affine space
Hypersurface
Concept in geometry and topology
§ Elliptic plane Hyperplane at infinity Parallel postulate Plane at infinity Point at infinity Weisstein, Eric W. "Line at Infinity". mathworld.wolfram
Line_at_infinity
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
Mathematical concept
In that case, the intersection point mentioned above lies on the hyperplane at infinity. If it is not improper, then it is proper. A proper sphere is elliptic
Affine_sphere
Coordinate system that is defined by points instead of vectors
complement of a hyperplane. The projective completion is unique up to an isomorphism. The hyperplane is called the hyperplane at infinity, and its points
Barycentric_coordinate_system
Type of geometry
affine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary". An algebraic model for doing
Projective_geometry
Completion of the usual space with "points at infinity"
points at infinity, which consists of homogenizing the defining polynomials, and removing the components that are contained in the hyperplane at infinity, by
Projective_space
Overview of and topical guide to geometry
Girard Desargues Desarguesian plane Line at infinity Point at infinity Plane at infinity Hyperplane at infinity Projective line Projective plane Oval (projective
Outline_of_geometry
Euclidean space without distance and angles
space that preserve affine space (equivalently, that leave the hyperplane at infinity invariant as a set) yield transformations of affine space. Conversely
Affine_space
Group of all affine transformations of an affine space
proceed from Pn to the affine space An by declaring a hyperplane ω to be a hyperplane at infinity, we obtain the affine group A {\displaystyle {\mathfrak
Affine_group
Research program on the symmetries of geometry
respecting (mapping to itself, not fixing pointwise) the chosen hyperplane at infinity. This subgroup has a known structure (semidirect product of the
Erlangen_program
Euclidean geometry without distance and angles
In projective geometry, affine space means the complement of a hyperplane at infinity in a projective space. Affine space can also be viewed as a vector
Affine_geometry
Mathematical theory
typical for an affine variety to acquire singular points on the hyperplane at infinity, when its closure in projective space is taken. Resolution says
Singularity_theory
Family of polynomials
subspaces of the space obtained by treating this fixed hyperplane as the hyperplane at infinity. Gaussian binomial coefficients play an important role
Gaussian_binomial_coefficient
Fiber bundle whose fibers are projective spaces
bundle (i.e., the structure sheaf). Then P(E) is a hyperplane in P(E ⊕ 1), called the hyperplane at infinity, and the complement of P(E) can be identified
Projective_bundle
Sphere-like surface
considered a projective ovoid in the projective closure (adding a hyperplane at infinity) of the affine space. O = { ( x 1 , . . . , x d ) ∈ R d | x 1 2
Ovoid_(projective_geometry)
Generalization of a vector bundle
complement is the open subscheme CR. The locus t = 0 is called the hyperplane at infinity. Let R be a quasi-coherent graded OX-algebra such that R0 = OX and
Cone_(algebraic_geometry)
Hypersurface in hyperbolic space
hyperplane at a given point, as their radii go towards infinity. In Euclidean geometry, such a "hypersphere of infinite radius" would be a hyperplane
Horosphere
Affine space over the complex numbers
complex projective space by fixing a hyperplane, which can be thought of as a hyperplane of ideal points "at infinity" of the affine space. To illustrate
Complex_affine_space
Projective plane Line at infinity Complex projective plane Complex projective space Plane at infinity, hyperplane at infinity Projective frame Projective
List of algebraic geometry topics
List_of_algebraic_geometry_topics
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
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
Representation of 3D and 4D polytopes
a facet. All vertices and edges of the polytope are projected onto a hyperplane of that facet. If the polytope is convex, a point near the facet will
Schlegel_diagram
Concept in projective geometry
pencil of hyperplanes in higher dimensions. A line segment on a projective line has as its dual the shape swept out by these lines or hyperplanes, a double
Duality_(projective_geometry)
Thing in mathematics and theoretical physics
point at which the pencil of lines normal to the tangent hyperplanes meet. If the quasi-sphere is a hyperplane, the centre is the point at infinity defined
Quasi-sphere
Isomorphism of projective spaces in geometry
Euclidean and affine spaces by the addition of new points called points at infinity. The term "projective transformation" originated in these abstract constructions
Homography
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
Branch of mathematics
such as convex bodies and normed spaces, as the dimension tends to infinity. It is at the intersection of convex geometry and functional analysis. The primary
Asymptotic_geometry
Number used in algebraic geometry
dimension n is the number of intersection points of the variety with n hyperplanes in general position. For an algebraic set, the intersection points must
Degree of an algebraic variety
Degree_of_an_algebraic_variety
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
Type of mathematical curve
of the cubic F ( X , Y , Z ) = 0. {\displaystyle F(X,Y,Z)=0.} A point at infinity of the cubic is a point such that Z = 0 {\displaystyle Z=0} . A real
Plane_cubic_curve
Type of geometric algebra
euclideanly orthogonal to (−1,a,b)—i.e., a plane; or in n dimensions, a hyperplane through the origin. This would cut another plane not through the origin
Conformal_geometric_algebra
Locus of the zeros of a polynomial of degree two
correspondence with the points of the quadric that do not belong to the tangent hyperplane at A. Expressing the points of the quadric in terms of the direction of
Quadric
Rational function of the form (az + b)/(cz + d)
Consider first the hyperplane in R4 given by x0 = 1. The celestial sphere may be identified with the sphere S+ of intersection of the hyperplane with the future
Möbius_transformation
Branch of number theory
space. It can be shown that the image of O× is a lattice that spans the hyperplane defined by x 1 + ⋯ + x r 1 + r 2 = 0. {\displaystyle x_{1}+\cdots +x_{r_{1}+r_{2}}=0
Algebraic_number_theory
Extending the elements of a polytope to form a new figure
stellation diagram of an n-polytope exists in an (n − 1)-dimensional hyperplane of a given facet. For example, in 4-space, the great grand stellated 120-cell
Stellation
Projection of a sphere through its center onto a plane
gnomonic projection can be taken of any n-dimensional hypersphere onto a hyperplane. The projection is the n-dimensional generalization of the trigonometric
Gnomonic_projection
Property of a mathematical space
tangent space at any Regular point of an algebraic variety. Another intuitive way is to define the dimension as the number of hyperplanes that are needed
Dimension
Geometric symmetry operation
reflections. More narrowly, a reflection refers to a reflection in a hyperplane ( n − 1 {\displaystyle n-1} dimensional affine subspace – a point on the
Point_reflection
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
not differentiable Supporting hyperplane - a hyperplane meeting certain conditions Supporting hyperplane theorem - that defines a supporting hyperplane
List_of_convexity_topics
Shading algorithm in computer graphics
{\displaystyle P_{H}} is the Householder matrix that reflects a point in the hyperplane that contains the origin and has the normal H . {\displaystyle H.} This
Blinn–Phong_reflection_model
Geometric figure
on the unit hyperbola. The conjugate diameter represents the spatial hyperplane of simultaneity corresponding to rapidity a. In this context the unit
Unit_hyperbola
projection of a variety into a hyperplane. They are so called because they appear to be singularities to an observer at the point being projected from
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Study of angle-preserving transformations
conformal geometry. The addition of a point at infinity to the space obviates the distinction between hyperplane and hypersphere; higher dimensional inversive
Inversive_geometry
Notation system for crystal lattice planes
means that the planes do not intersect that axis (the intercept is "at infinity"). Considering only (hkl) planes intersecting one or more lattice points
Miller_index
Multivariate statistical technique
\Phi :\mathbb {R} ^{d}\to \mathbb {R} ^{N}} , it is easy to construct a hyperplane that divides the points into arbitrary clusters. Of course, this Φ {\displaystyle
Kernel principal component analysis
Kernel_principal_component_analysis
All points whose relative distances to two circles are same
hyperspheres in Euclidean space of any dimension, giving the radical hyperplane of two non-concentric hyperspheres. Michel Chasles, C. H. Schnuse: Die
Radical_axis
Generalized sphere of dimension n (mathematics)
-sphere onto n {\displaystyle n} -space with a single adjoined point at infinity; under the metric thereby defined, R n ∪ { ∞ } {\displaystyle \mathbb
N-sphere
Non-orientable surface with one edge
strip, one that is fully four-dimensional and for which all cuts by hyperplanes separate it into two parts that are topologically equivalent to disks
Möbius_strip
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
4-dimensional object
Then roll the cylinder in the plane perpendicular to the 3-dimensional hyperplane that the cylinder lies in, so that its two circular ends meet. The resulting
Duocylinder
Mathematical theorem
\{(s_{1},\ldots ,s_{n})\in I_{s,t}^{n}\mid s_{i}=s_{j}\}} is contained in a hyperplane, hence by an application of Fubini's theorem its measure with respect
Grönwall's_inequality
Four-dimensional number system
Dublin, when Hamilton was on his way to the Royal Irish Academy to preside at a council meeting. As he walked along the towpath of the Royal Canal with
Quaternion
Curve defined as zeros of polynomials
0}(\mathbb {P} ^{2},d\cdot [H])} (where [ H ] {\displaystyle [H]} is the hyperplane class) parametrizing all such stable curves. A dimension count can be
Algebraic_curve
Australian and American mathematician (born 1975)
singular integral operators with the multiplier allowed to degenerate on a hyperplane, identifying conditions which ensure operator continuity relative to Lp
Terence_Tao
Concept in algebraic geometry
H ] {\displaystyle K=(d-3)[C\cap H]} where H {\displaystyle H} is the hyperplane divisor. First work in the affine chart Z ≠ 0 {\displaystyle Z\neq 0}
Adjunction_formula
Result in general relativity
{\displaystyle h_{ab}=g_{ab}+X_{a}\,X_{b}} is the projection tensor onto the hyperplanes orthogonal to X → {\displaystyle {\vec {X}}} . Also, dot denotes differentiation
Raychaudhuri_equation
points in Pn in linear general position (that is, with no n + 1 lying in a hyperplane), there is a unique rational normal curve passing through them. The curve
Rational_normal_curve
American mathematician (born 1949)
MR 1362980. Charney, Ruth; Davis, Michael W. (1995). "The K(π,1)-problem for hyperplane complements associated to infinite reflections groups". Journal of the
Michael_W._Davis
Five Islands theorem, Cartan's theorem on holomorphic curves omitting hyperplanes, Hayman's result that an exceptional set of radii is unavoidable in Nevanlinna
Bloch's_principle
Particular mapping that projects a sphere onto a plane
+ 1)-dimensional Euclidean space En+1. If Q is a point of Sn and E a hyperplane in En+1, then the stereographic projection of a point P ∈ Sn − {Q} is
Stereographic_projection
Generalization of the tangent space to a manifold to the case of certain spaces
containing K {\displaystyle K} and bounded by the supporting hyperplanes of K {\displaystyle K} at x {\displaystyle x} . The boundary T K {\displaystyle T_{K}}
Tangent_cone
Machine learning kernel function
to zero with increasing d, whereas when xTy + c > 1, K(x, y) tends to infinity. Yoav Goldberg and Michael Elhadad (2008). splitSVM: Fast, Space-Efficient
Polynomial_kernel
Mathematical model combining space and time
kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich
Spacetime
Model of hyperbolic geometry
defining a point in the hyperboloid model, we may project it onto the hyperplane t = 0 by intersecting it with a line drawn through [−1, 0, ..., 0]. The
Poincaré_disk_model
Mathematical problem
necklace is chosen uniformly at random from the set of necklaces with t colors and m beads of each color. As m tends to infinity, the probability that the
Necklace_splitting_problem
Geometry founded on spheres
point in n-dimensional space (which may be the point at infinity) together with an oriented hyperplane passing through that point. The space Z 2n − 1 is
Lie_sphere_geometry
number projective line, which adjoins to the dual numbers a set of points at infinity. Topologically, this projective line is equivalent to a cylinder. Points
Laguerre_transformations
In projective geometry, points that define coordinates
that no hyperplane contains n + 1 of them. A projective frame is sometimes called a simplex, although a simplex in a space of dimension n has at most n
Projective_frame
Schlumprecht (Schlumprecht (1991)), on which depend Gowers' solution to Banach's hyperplane problem and the Odell–Schlumprecht solution to the distortion problem
Tsirelson_space
Matroid without short circuits
are the hyperplanes of a paving matroid on S {\displaystyle S} . If a paving matroid has rank d + 1 {\displaystyle d+1} , then its hyperplanes form a set
Paving_matroid
Indicator for how well data points fit a line or curve
worst possible least-squares predictor (equivalent to a horizontal hyperplane at a height equal to the mean of the observed data). This occurs when a
Coefficient_of_determination
Function of the coefficients of a polynomial that gives information on its roots
of the points of V (including the points at infinity), which either are singular or have a tangent hyperplane that is parallel to the axis of the selected
Discriminant
Polyhedron associated with another by swapping vertices for faces
extended Euclidean space, may be formed by adding the required 'plane at infinity'. Some theorists prefer to stick to Euclidean space and say that there
Dual_polyhedron
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
Fundamental object of geometry
Foundations of geometry Locus (mathematics) Position (geometry) Point at infinity Point cloud Point process Point set registration Pointwise Singular point
Point_(geometry)
Geometric space with four dimensions
link from Internet Archive. Gamow, George (1988). One Two Three . . . Infinity: Facts and Speculations of Science (3rd ed.). Courier Dover Publications
Four-dimensional_space
Subdivision of the plane by lines
source gives a formula for the number of cells of variable dimension in a hyperplane arrangement of variable dimension, which simplifies to ( n 2 ) = n ( n
Arrangement_of_lines
Type of fair division
respect to their measure, i.e. volume) with a single (n − 1)-dimensional hyperplane. Stated differently: if the cake is the space R n {\displaystyle \mathbb
Consensus_splitting
Statistical model validation technique
.., xip. If least squares is used to fit a function in the form of a hyperplane ŷ = a + βTx to the data (xi, yi) 1 ≤ i ≤ n, then the fit can be assessed
Cross-validation_(statistics)
Middle quantile of a data set or probability distribution
such that any hyperplane that goes through that point divides the set of points in two roughly equal subsets: the smaller part should have at least a 1/(d + 1)
Median
Tree-based ensemble machine learning methods
in 1995. Ho established that forests of trees splitting with oblique hyperplanes can gain accuracy as they grow without suffering from overtraining, as
Random_forest
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
Theorem in geometry
∈ l {\textstyle t\in l} let H t {\textstyle H_{t}} denote the affine hyperplane orthogonal to l {\textstyle l} that passes through t {\textstyle t} .
Brunn–Minkowski_theorem
Generalized function whose value is zero everywhere except at zero
transform because it recovers the value of φ(x) from its integrals over hyperplanes. For instance, if n is odd and k = 1, then the integral on the right
Dirac_delta_function
number of components in a partitioning of an n-dimensional space by m hyperplanes. 4. The domain is the real line R {\displaystyle \mathbb {R} } . The
Growth_function
Mathematical metric in geometry
this way. Cayley–Klein Voronoi diagrams are affine diagrams with linear hyperplane bisectors. Cayley–Klein metric is first illustrated on the real projective
Cayley–Klein_metric
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
List of concepts in artificial intelligence
of choosing a set of optimal hyperparameters for a learning algorithm. hyperplane A decision boundary in machine learning classifiers that partitions the
Glossary of artificial intelligence
Glossary_of_artificial_intelligence
Type of non-Euclidean geometry
intersections with planes through the origin, dihedral angles between hyperplanes can be described by inner products of normal vectors, and hyperbolic
Hyperbolic_geometry
Element of an exterior algebra
intersection of the lines through the origin of Rn+1 with a selected hyperplane, such as H: xn+1 = 1. Lines through the origin of R3 intersect the plane
Multivector
In functional analysis, a Hilbert space
products of elements in the feature space and can accordingly be seen as hyperplanes. This view of the RKHS is related to the kernel trick in machine learning
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
Maximally symmetric Lorentzian manifold with a negative cosmological constant
half-space coordinates and it is bounded by two null, aka lightlike, geodesic hyperplanes; the green shaded area on the surface corresponds to the region of conformal
Anti-de_Sitter_space
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HYPERPLANE AT-INFINITY
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