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Affine space over the complex numbers
linear maps." Accordingly, a complex affine space, that is an affine space over the complex numbers, is like a complex vector space, but without a distinguished
Complex_affine_space
Euclidean space without distance and angles
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent
Affine_space
Index of articles associated with the same name
A complex space is a mathematical space based upon complex numbers. Types of complex space include: Complex affine space, an affine space over the complex
Complex_space
Space formed by the ''n''-tuples of complex numbers
coordinate systems on complex manifolds. Complex affine space Coordinate space Gunning, Robert; Hugo Rossi, Analytic functions of several complex variables
Complex_coordinate_space
Fundamental space of geometry
not distinct) in the complex affine space. Therefore, most of algebraic geometry is built in complex affine spaces and affine spaces over algebraically
Euclidean_space
Generalization of a complex manifold that allows the use of singularities
{\mathbb {C} }}} . Choose an open subset U {\displaystyle U} of some complex affine space C n {\displaystyle \mathbb {C} ^{n}} , and fix finitely many holomorphic
Complex_analytic_variety
Real affine space of even dimension that is not isomorphic to a complex affine space
In algebraic geometry, an exotic affine space is a complex algebraic variety that is diffeomorphic to R 2 n {\displaystyle \mathbb {R} ^{2n}} for some
Exotic_affine_space
Analytic space in mathematics
Tate's notion of a rigid analytic space. In the complex case, algebraic geometry begins by defining the complex affine space to be C n . {\displaystyle \mathbb
Berkovich_space
Manifold or algebraic variety of dimension n in a space of dimension n+1
which is embedded in an ambient space of dimension n, generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces
Hypersurface
Mathematical space with two coordinates
Some two-dimensional mathematical spaces are not used to represent physical positions, like an affine plane or complex plane. The most basic example is
Two-dimensional_space
Group of all affine transformations of an affine space
mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself
Affine_group
Mathematical concept
inequality for complex projective space Projective Hilbert space Quaternionic projective space Real projective space Complex affine space K3 surface Besse
Complex_projective_space
Partition of space by hyperplanes
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
Arrangement_of_hyperplanes
Algebraic variety defined within an affine space
geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally
Affine_variety
Algebraic structure in linear algebra
Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars
Vector_space
Construct allowing differentiation of tangent vector fields of manifolds
differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector
Affine_connection
Mathematical set with some added structure
real, complex, and more generally, over any field. Every real or complex affine or projective space is also a topological space. An affine space is a non-compact
Space_(mathematics)
Theorem in algebraic geometry
Abhyankar–Moh theorem states that if L {\displaystyle L} is a complex line in the complex affine plane C 2 {\displaystyle \mathbb {C} ^{2}} , then every embedding
Abhyankar–Moh_theorem
Mathematical object studied in the field of algebraic geometry
an affine algebraic variety. Let k = C, and A2 be the two-dimensional affine space over C. Polynomials in the ring C[x, y] can be viewed as complex valued
Algebraic_variety
Generalization of a scheme
given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer
Algebraic_space
Class of mathematical function
z_{2})=z_{1}/z_{2}} is a meromorphic function on the two-dimensional complex affine space. Here it is no longer true that every meromorphic function can be
Meromorphic_function
2D surface which extends indefinitely
which is homeomorphic to an open disk. Viewing the plane as an affine space produces the affine plane, which lacks a notion of distance but preserves the notion
Plane_(mathematics)
In algebraic geometry, the Koras–Russell cubic threefolds are smooth affine complex threefolds diffeomorphic to C 3 {\displaystyle \mathbf {C} ^{3}} studied
Koras–Russell_cubic_threefold
Euclidean geometry without distance and angles
parallelism of lines. Affine geometry can be developed in two ways that are essentially equivalent. In synthetic geometry, an affine space is a set of points
Affine_geometry
Study of complex manifolds and several complex variables
contrast to complex manifolds which are always smooth, complex geometry is also concerned with possibly singular spaces. An affine complex analytic variety
Complex_geometry
Type of Kac–Moody algebras
affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie
Affine_Lie_algebra
Completion of the usual space with "points at infinity"
space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one
Projective_space
Vector space with generalized dot product
product space is a real or complex vector space endowed with an operation called an inner product. The inner product of two vectors in the space is a scalar
Inner_product_space
Set of a ring's prime ideals
and the associated ringed space is called an affine scheme. The spectrum of a ring R {\displaystyle R} and the associated affine scheme are both denoted
Spectrum_of_a_ring
Concept in geometry and topology
Riemann sphere, which is therefore a 2-sphere, being added to a complex affine space of two dimensions over C {\displaystyle \mathbb {C} } (so four real
Line_at_infinity
Siegel domain or Piatetski-Shapiro domain is a special open subset of complex affine space generalizing the Siegel upper half plane studied by Siegel (1939)
Siegel_domain
Space formed by the ''n''-tuples of real numbers
vector space. It is a Euclidean space and a real affine space, and every Euclidean or affine space is isomorphic to it. It is an analytic manifold, and
Real_coordinate_space
Method of defining surface detail on a computer-generated graphic or 3D model
triangles for rendering and affine mapping is used on them. The reason this technique works is that the distortion of affine mapping becomes much less noticeable
Texture_mapping
Geometric representation of the complex numbers
view of the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where the inner product of complex numbers w
Complex_plane
Number line and triangular tiling's symmetry mathematical structure
interpretation. The affine symmetric groups have close relationships with other mathematical objects, including juggling patterns and certain complex reflection
Affine_symmetric_group
Generalization of algebraic variety
nonempty topological space.) Generic point. The points of the affine line A1 C, as a scheme, are its complex points (one for each complex number) together
Scheme_(mathematics)
Field of mathematics which studies incidence structures
geometric examples, particularly projective planes and affine planes. A projective plane is a linear space in which: Every pair of distinct lines meet in exactly
Incidence_geometry
Geometric model of the physical space
model physical space as a three-dimensional affine space E ( 3 ) {\displaystyle E(3)} over the real numbers. This is unique up to affine isomorphism. It
Three-dimensional_space
(pseudo-)Riemannian manifold whose geodesics are reversible
M = G / H is a symmetric space, then Nomizu showed that there is a G-invariant torsion-free affine connection (i.e. an affine connection whose torsion
Symmetric_space
Mathematical group occurring in algebraic geometry and the theory of complex manifolds
of the affine line with two origins over k is isomorphic to Z. The Picard group of the n {\displaystyle n} -dimensional complex affine space: Pic (
Picard_group
Algebraic topology theory
dimension reason). Ω {\displaystyle \Omega } is an infinite-dimensional complex affine space and is therefore contractible. Let G {\displaystyle {\mathcal {G}}}
Equivariant_cohomology
Branch of mathematics
always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed. We consider the affine space of dimension
Algebraic_geometry
finite index. An affine complex manifold is a complex manifold that has an atlas whose transition maps belong to the group of complex affine transformations
Affine_manifold
Topics referred to by the same term
coordinate space Real manifold Real vector space Real affine space Real spaces can also mean: The book Real Spaces: World Art History and the Rise of Western
Real_space
Framework for multi-scale signal representation
"Generalized Gaussian Scale-Space Axiomatics Comprising Linear Scale-Space, Affine Scale-Space and Spatio-Temporal Scale-Space". Journal of Mathematical
Scale_space
Topology on prime ideals and algebraic varieties
algebraic geometry, k is usually the field of complex numbers). First, we define the topology on the affine space A n {\displaystyle \mathbb {A} ^{n}} , formed
Zariski_topology
Manifold
manifolds including, for example, smooth complex affine algebraic varieties. The classification of complex manifolds is much more subtle than that of
Complex_manifold
Coordinate system using perpendicular axes
and the spherical and cylindrical coordinates for three-dimensional space. An affine line with a chosen Cartesian coordinate system is called a number line
Cartesian_coordinate_system
Sum of terms, each multiplied with a scalar
defined as subsets of an ambient vector space (except for affine spaces, which are also considered as "vector spaces forgetting the origin"), rather than
Linear_combination
Curve defined as zeros of polynomials
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in
Algebraic_curve
Mathematical structure
Weyl group, the Coxeter complex is a subdivision of the affine plane and one speaks of affine, or Euclidean, buildings. An affine building of type Ã1 is
Building_(mathematics)
Concept in geometry
Projective spaces Pn for n > 1 are not one-point compactifications of corresponding affine spaces for the reason mentioned above under § Affine geometry
Point_at_infinity
Term in mathematics
Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes
Stein_manifold
Open problem in convex geometry
In mathematics, the affine plank problem is an open question in convex geometry posed by Thøger Bang in 1951 as a strengthening of Tarski's plank problem
Affine_plank_problem
is complete (i.e., affinely diffeomorphic to a quotient space of the affine space under a proper action of a discrete group of affine transformations, then
Chern's conjecture (affine geometry)
Chern's_conjecture_(affine_geometry)
American mathematician
Fried on affine structures on manifolds, and work in real projective structures on compact surfaces. In particular he proved that the space of convex
William Goldman (mathematician)
William_Goldman_(mathematician)
Number with a real and an imaginary part
triangle will remain the same, when the complex plane is transformed by translation or dilation (by an affine transformation), corresponding to the intuitive
Complex_number
Topological space of dimension zero
In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several
Zero-dimensional_space
Steiner surface, a realization of the real projective plane in real affine space Tori, surfaces of revolution generated by a circle about a coplanar axis
List of complex and algebraic surfaces
List_of_complex_and_algebraic_surfaces
Branch of mathematics
an affine space, where collinearity and ratios can be studied but not distances; it can be studied as the complex plane using techniques of complex analysis;
Geometry
Feature detection algorithm in computer vision
scaling, orientation, illumination changes, and partially invariant to affine distortion. This section summarizes the original SIFT algorithm and mentions
Scale-invariant feature transform
Scale-invariant_feature_transform
Locus of the zeros of a polynomial of degree two
quadric is an affine algebraic variety, or, if it is reducible, an affine algebraic set. Quadrics may also be defined in projective spaces; see § Normal
Quadric
Mathematical model combining space and time
In physics, spacetime, or the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into
Spacetime
Coordinate system that is defined by points instead of vectors
Euclidean space, a flat or an affine space A {\displaystyle \mathbf {A} } of dimension n that are affinely independent; this means that there is no affine subspace
Barycentric_coordinate_system
Mathematical space
{Gr} _{k}(V)\,|\,\dim(W\cap V_{n-k+j-\lambda _{j}})=j\}.} These are affine spaces, and their closures (within the Zariski topology) are Schubert varieties
Grassmannian
Geometric model of the planar projection of the physical universe
numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has also
Euclidean_plane
Complex numbers with non-negative imaginary part
Half-planes are an example of two-dimensional half-space. A half-plane can be split in two quadrants. The affine transformations of the upper half-plane include
Upper_half-plane
Multi-dimensional generalization of triangle
the k + 1 points u 0 , … , u k {\displaystyle u_{0},\dots ,u_{k}} are affinely independent, which means that the k vectors u 1 − u 0 , … , u k − u 0 {\displaystyle
Simplex
Theorem about the dual of a Hilbert space
isometry) complex Hilbert space, called its complexification, which is why Hilbert spaces are often automatically assumed to be complex. Real and complex Hilbert
Riesz_representation_theorem
Property of a mathematical space
Euclidean space is defined. While analysis usually assumes a manifold to be over the real numbers, it is sometimes useful in the study of complex manifolds
Dimension
Symmetry group of a configuration in space
faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations
Space_group
Concept in differential geometry
in the case of complex affine holonomies, as demonstrated by Schwachhöfer (2001). Let V be a finite-dimensional complex vector space, let H ⊂ Aut(V)
Holonomy
In geometry, set whose intersection with every line is a single line segment
functions is called convex analysis. Spaces in which convex sets are defined include the Euclidean spaces, the affine spaces over the real numbers, and certain
Convex_set
Concept in mathematics
also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also
Morphism of algebraic varieties
Morphism_of_algebraic_varieties
Algebraic variety containing an algebraic torus
examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space. A precise definition is that
Toric_variety
Set of vectors used to define coordinates
convex set, and cone have related notions of basis. An affine basis for an n-dimensional affine space is n + 1 {\displaystyle n+1} points in general linear
Basis_(linear_algebra)
which admits an open affine covering { S p e c ( A i ) → X } {\displaystyle \{Spec(A_{i})\to X\}} . From the locally ringed space point-of-view, a derived
Derived_scheme
Geometric system with a finite number of points
finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. Other significant types
Finite_geometry
Algebraic variety in a projective space
structure is as follows. The projective space P n {\displaystyle \mathbb {P} ^{n}} is covered by the standard open affine charts U i = { [ x 0 : ⋯ : x n ]
Projective_variety
Concept in algebraic geometry
smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of
Smooth_scheme
Space with one dimension
is a one-dimensional space. In particular, if the field is the complex numbers C , {\displaystyle \mathbb {C} ,} then the complex projective line P 1 (
One-dimensional_space
Analogue of a complex analytic space over a nonarchimedean field
euclidean space, or schemes being coverable by affines. Schemes over k can be analytified functorially, much like varieties over the complex numbers can
Rigid_analytic_space
a complex line is a one-dimensional affine subspace of a vector space over the complex numbers. A common point of confusion is that while a complex line
Complex_line
Straight path on a curved surface or a Riemannian manifold
having vanishing geodesic curvature. More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain
Geodesic
the ovoid becomes an affine ovoid in the affine space corresponding to ε ∞ {\displaystyle \varepsilon _{\infty }} . Also, any affine ovoid can be considered
Ovoid_(projective_geometry)
Parametrizes complex structures on a surface
Teichmüller space T ( S ) {\displaystyle T(S)} of a (real) topological (or differential) surface S {\displaystyle S} is a space that parametrizes complex structures
Teichmüller_space
Number of vectors in any basis of the vector space
field F . {\displaystyle F.} The complex numbers C {\displaystyle \mathbb {C} } are both a real and complex vector space. Their dimension depends on the
Dimension_(vector_space)
Four-dimensional number system
the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space. Points
Quaternion
Research program on the symmetries of geometry
n-dimensional real projective space (the general linear group of degree n + 1, quotiented by scalar matrices). The affine group will be the subgroup respecting
Erlangen_program
G being a Lie group and X a homogeneous space for G. Foundational examples are hyperbolic manifolds and affine manifolds. Let X {\displaystyle X} be a
(G,_X)-manifold
Number of intersection points of algebraic curves and hypersurfaces
of the possibility of a finite number of intersection points in the affine space, together with infinitely many intersection points at infinity. This
Bézout's_theorem
Point not between two other points
extreme point of a convex set S {\displaystyle S} in a real or complex vector space or affine space is a point in S {\displaystyle S} that does not lie in any
Extreme_point
Lie groups and their associated Lie algebras
C. Note that every complex Lie algebra can also be viewed as a real Lie algebra of twice the dimension. The Lie algebra of affine transformations of dimension
Table_of_Lie_groups
affine normal vector field, or the Blaschke normal field. A special (i.e. det = 1) affine transformation of real (n + 1)-space will carry the affine normal
Affine_focal_set
Branch of mathematics
to relate the tangent spaces at different points, i.e. a notion of parallel transport. An important example is provided by affine connections. For a surface
Differential_geometry
Group that admits a formal description in terms of reflections
translates of these hyperplanes. The affine Coxeter group (or affine Weyl group) is then the group generated by the (affine) reflections about all the hyperplanes
Coxeter_group
Geometric space with four dimensions
Four-dimensional (4D) space is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible
Four-dimensional_space
numbers. The usage of complex numbers (instead of the real numbers) does not lead to a Möbius plane, because in the complex affine plane the curve x 2 +
Möbius_plane
Quadratic polynomial
{\displaystyle f_{c}(x)} is affine conjugate to the general form of the quadratic polynomial it is often used to study complex dynamics and to create images
Complex_quadratic_polynomial
Italian-born American mathematician (1923–2023)
moduli space of space forms, a characterization of when a metric can be found so that a given differential form is harmonic, and various works on affine geometry
Eugenio_Calabi
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