We can use the standard ideas in singularity theory to classify, up to local diffeomorphism, the affine focal set.If the family of affine distance functions can be shown to be a certain kind of family then the local structure is known.We want the family of affine distance functions to be a versal unfolding of the singularities which arrise.…

A scheme is a locally ringed space X admitting a covering by open sets U i, such that the restriction of the structure sheaf O X to each U i is an affine scheme.Therefore one may think of a scheme as being covered by "coordinate charts" of affine schemes.The above formal definition means exactly that schemes are obtained by glueing together affine schemes for the Zariski topology.…

They are also useful for the study of affine curves.

A fundamental group of a compact complete flat affine manifold is called an affine crystallographic group.Classification of affine crystallographic groups is a difficult problem, far from being solved.The Riemannian crystallographic groups (also known as Bieberbach groups) were classified by Ludwig Bieberbach, answering a question posed by David Hilbert.…

According to another basic theorem, any group in the category of affine varieties has a faithful finite-dimensional linear representation: we can consider it to be a matrix group over K, defined by polynomials over K and with matrix multiplication as the group operation.For that reason a concept of affine algebraic group is redundant over a field -- we may as well use a very concrete definition.…

In the case of affine Coxeter groups like, or, one mirror, usually the last, is translated off the origin.

Tits later reworked the foundational aspects of the theory of buildings using the notion of a chamber system, encoding the building solely in terms of adjacency properties of simplices of maximal dimension; this leads to simplifications in both spherical and affine cases.He proved that, in analogy with the spherical case, every building of affine type and rank at least four arises from a group.…

Affine connections can be defined within Cartan's general framework.In the modern approach, this is closely related to the definition of affine connections on the frame bundle.Indeed, in one formulation, a Cartan connection is an absolute parallelism of a principal bundle satisfying suitable properties.…

Quantitative analysis of affine region detectors take into account both the accuracy of point locations and the overlap of regions across two images.

The various types of affine geometry correspond to what interpretation is taken for rotation.Euclidean geometry corresponds to the ordinary idea of rotation, while Minkowski's geometry corresponds to hyperbolic rotation.…

Positions and displacements all have units of length, but their meaning is not interchangeable: This illustrates the subtle distinction between affine quantities (ones modeled by an affine space, such as position) and vector quantities (ones modeled by a vector space, such as displacement).Properly then, positions have dimension of affine length, while displacements have dimension of vector length.To assign a number to an affine unit, one must not only choose a unit of measurement, but also a point of reference, while to assign a number to a vector unit only requires a unit of measurement.…

Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading to Artin stacks and, even finer, Deligne-Mumford stacks, both often called algebraic stacks.Sometimes other algebraic sites replace the category of affine schemes.For example, Nikolai Durov has introduced commutative algebraic monads as a generalization of local objects in a generalized algebraic geometry.…

For any reasonable property, it turns out that the property is true generically on the subvariety (in the sense of being true on an open dense subset) if and only if the property is true at the generic point.Such results are frequently proved using the methods of limits of affine schemes developed in EGA IV 8.A related concept in algebraic geometry is general position, whose precise meaning depends on the context.…

The following proof uses only notions of affine geometry, notably homothecies.

In geometry, an affine plane is a two-dimensional affine space.Typical examples of affine planes are All the affine planes defined over a field are isomorphic.…

Another technique was subdividing the polygons into smaller polygons, like triangles in 3d-space or squares in screen space, and using an affine mapping on them.The distortion of affine mapping becomes much less noticeable on smaller polygons.Yet another technique was approximating the perspective with a faster calculation, such as a polynomial.…

The dual function g is concave, even when the initial problem is not convex, because it is a point-wise infimum of affine functions.

Since the sphere is not an affine space, familiar properties of affine constructions may fail, though the constructed curves may otherwise be entirely satisfactory.For example, the de Casteljau algorithm may be used to split a curve in affine space; this does not work on a sphere.…

Affine space is a geometry in this sense, and is equipped with a flat Cartan connection.Thus a general affine manifold is viewed as curved deformation of the flat model geometry of affine space.Informally, an affine space is a vector space without a fixed choice of origin.…

Examples with more than two variables include characters of some irreducible highest-weight representations of affine Kac-Moody algebras.Meromorphic Jacobi forms appear in the theory of Mock modular forms.…