The inverse cross ratio is invariant under projective transformations and thus makes sense for points in the projective line.However, the formula above only makes sense for points in the affine line.In the slightly more general set-up below, the cross ratio makes sense for any four collinear points in projective space One just identifies the line containing the points with the projective line by a suitable projective transformation and then uses the formula above.…

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.…

This meant the walls would be a constant distance along a vertical line and the floors/ceilings would be a constant distance along a horizontal line.A fast affine mapping could be used along those lines because it would be correct.…

Affine types are a version of linear types imposing weaker constraints, corresponding to affine logic.An affine resource can only be used once, while a linear one must be used once.Relevant types correspond to relevant logic which allows exchange and contraction but not weakening, which translates to every variable being used at least once.…

In mathematics, an affine space is a geometric structure that generalizes certain properties of parallel lines in Euclidean space.In an affine space, there is no distinguished point that serves as an origin.Hence, no vector has a fixed origin and no vector can be uniquely associated to a point.…

This is a group under the operation of composition of functions, called the affine group.Ordinary matrix-vector multiplication always maps the origin to the origin, and could therefore never represent a translation, in which the origin must necessarily be mapped to some other point.…

In this case there are three different buildings, two spherical and one affine.Each is a union of apartments, themselves simplicial complexes.…

From an economic anthropology viewpoint, the products of fan labor are a form of cultural wealth, valuable also for their ability to interrelate the fan works, the fan-creators, and the original media property itself through conversation and fan work exchanges.Fans, in other words, are "affines" of media property and of other fans.From another economic anthropology perspective, fan creative practices are labor that is done in a relatively routine way and that helps to maintain a connection to the media property itself (the "cultural ancestor" or "deity").…

In this language, an affine connection is simply a covariant derivative or (linear) connection on the tangent bundle.However, this approach does not explain the geometry behind affine connections nor how they acquired their name.…

A fractional program in which f is nonnegative and concave, g is positive and convex, and S is a convex set is called a concave fractional program.If g is affine, f does not have to be restricted in sign.…

One can check that any affine involution in fact has this form.Geometrically this means that any affine involution can be obtained by taking oblique reflections against any number from 0 through n hyperplanes going through a point b.…

These last two points are quite hard to make precise, so affine connections are more often defined infinitesimally.

In Euclidean geometry, oblique reflections generalize ordinary reflections by not requiring that reflection be done using perpendiculars.If two points are oblique reflections of each other, they will still stay so under affine transformations.Consider a plane P in the three-dimensional Euclidean space.…

The affine subspaces are model surfaces -- they are the simplest surfaces in R 3, and are homogeneous under the Euclidean group of the plane, hence they are Klein geometries in the sense of Felix Klein's Erlangen programme.Every smooth surface S has a unique affine plane tangent to it at each point.The family of all such planes in R 3, one attached to each point of S, is called the congruence of tangent planes.…

More precisely, an affine space is a set with a free transitive vector space action.In particular, a vector space is an affine space over itself, by the map.…

In that case, the intersection point mentioned above lies on the hyperplane at infinity.Affine spheres have been the subject of much investigation, with many hundreds of research articles devoted to their study.…

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space.More generally, a half-space is either of the two parts into which a hyperplane divides an affine space.That is, the points that are not incident to the hyperplane are partitioned into two convex sets (i.e., half-spaces), such that any subspace connecting a point in one set to a point in the other must intersect the hyperplane.…

Here the word 'random' must be interpreted as subject to correct symmetry considerations.There is a sample space of lines, one on which the affine group of the plane acts.A probability measure is sought on this space, invariant under the symmetry group.…

In 2004 Hamburg Records, based at the world famous red-light district St. Pauli Reeperbahn, started their mail-order with just management and label attached band products.Throughout the years this changed and the company runs that business for quite a few namable bands and music-affine brands.Since 2006 Hamburg Records offers with Shirtagentur the service to print all kinds of merchandising and grew within just a few years to one of the biggest suppliers, especially for all kinds of Punk, Rock and Metal bands in Central Europe.…

For the mentioned \alpha -families the affine connection is called the \alpha -connection and can also be expressed in more ways.