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 transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space...

equipped with an inner product. The action of translations makes the space an affine space, and this allows defining lines, planes, subspaces, dimension, and...

differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector...

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...

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...

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...

affine space is a vector space that's forgotten its origin". In particular, every linear space is also an affine space. Given an n-dimensional affine...

faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations...

bioinformatics Affine geometry, a geometry characterized by parallel lines Affine group, the group of all invertible affine transformations from any affine space over...

half-space can be either open or closed. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space...

{\displaystyle \mathbb {R} } for a Euclidean space), and the affine combination is also a point. See Affine space § Affine combinations and barycenter for the...

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...

In other kinds of ambient spaces, some properties from Euclidean space are no longer relevant. For example, in affine space, there is no concept of distance...

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...

metrical notions of distance or angle are. Affine spaces differ from linear spaces (that is, vector spaces) in that they do not have a distinguished choice...

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...

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...

] . {\displaystyle k[X_{1},\ldots ,X_{n}].} An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by...

Roughly, affine spaces are vector spaces whose origins are not specified. More precisely, an affine space is a set with a free transitive vector space action...

Oriented hyperbolic space: SO+(1, n) / SO(n) Anti-de Sitter space: AdSn+1 = O(2, n) / O(1, n) Others Affine space over field K (for affine group, point stabilizer...

structure is called a ringed space or a sheaf of rings. Formally, a scheme is a ringed space covered by affine schemes. An affine scheme is the spectrum of...

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...

equations. Some mathematical spaces have additional arithmetical structure associated with their points. A vector plane is an affine plane whose points, called...

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...

algebra Clifford algebra Geometric algebra Affine space Affine transformation Affine group Affine geometry Affine coordinate system Flat (geometry) Cartesian...

natural number n, let An be an affine n-space over K, identified to K n {\displaystyle K^{n}} through the choice of an affine coordinate system. The polynomials...

true if we assume only that k is algebraically closed. We consider the affine space of dimension n over k, denoted An(k) (or more simply An, when k is clear...

subscheme of some projective space. An affine space is a Zariski-open subset of a projective space, and since any closed affine subset U {\displaystyle U}...

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...