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.…

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 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.…

Hence, no vector has a fixed origin and no vector can be uniquely associated to a point.In an affine space, there are instead displacement vectors between two points of the space.Thus it makes sense to subtract two points of the space, giving a vector, but it does not make sense to add two points of the space.…

Also, sets of parallel lines remain parallel after an affine transformation.An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.…

However these affine planes all have a marked point, the point of contact with the surface, and they are tangent to the surface at this point.The confusion therefore arises because an affine space with a marked point can be identified with its tangent space at that point.…

A straight line in the affine space is, by definition, its intersection with a two-dimensional linear subspace (plane through the origin) of the (n+1) -dimensional linear space.Every linear space is also an affine space.Every point of the affine space is its intersection with a one-dimensional linear subspace (line through the origin) of the (n+1) -dimensional linear space.…

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.It describes the geometry of points and free vectors in space.…

Methods of drawing an ellipse usually require the axes of the ellipse to be known.An ellipse can be seen as an image of the unit circle under an affine transformation.…

Any anisotropic or isotropic linear algebraic group over a field becomes split over the algebraic closure, so this distinction is interesting from the point of view of Algebraic number theory.Let G be a unipotent group acting on an affine variety.Then every G-orbit in the variety is closed.…

In a third paper, Shi and Tomasi proposed an additional stage of verifying that features were tracked correctly.An affine transformation is fit between the image of the currently tracked feature and its image from a non-consecutive previous frame.If the affine compensated image is too dissimilar the feature is dropped.…

The first one, successfully tackled by Hilbert in the case of a general linear group, is to prove that the algebra A is finitely generated.This is necessary if one wanted the quotient to be an affine algebraic variety.Whether a similar fact holds for arbitrary groups G was the subject of Hilbert's fourteenth problem, and Nagata demonstrated that the answer was negative in general.…

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.In particular, a vector space is an affine space over itself, by the map.…

In geometry, any hyperplane H of a projective space P may be taken as a hyperplane at infinity.Then the set complement is called an affine space.For instance, if are homogeneous coordinates for n-dimensional projective space, then the equation defines a hyperplane at infinity for the n-dimensional affine space with coordinates, H is also called the ideal hyperplane.…

If we call both circles and such completed lines cycles, we get an incidence structure in which every three points determine exactly one cycle.In an affine plane the parallel relation between lines is essential.In the geometry of cycles, this relation is generalized to the touching relation.…

A fixed point of an isometry group is a point that is a fixed point for every isometry in the group.For any isometry group in Euclidean space the set of fixed points is either empty or an affine space.For an object, any unique center and, more generally, any point with unique properties with respect to the object is a fixed point of its symmetry group.…

Under repeated iteration of the horseshoe map, most orbits end up at the fixed point in the left cap.This is because the horseshoe maps the left cap into itself by an affine transformation that has exactly one fixed point.Any orbit that lands on the left cap never leaves it and converges to the fixed point in the left cap under iteration.…

For instance, if are homogeneous coordinates for n-dimensional projective space, then the equation defines a hyperplane at infinity for the n-dimensional affine space with coordinates, H is also called the ideal hyperplane.Similarly, starting from an affine space A, every class of parallel lines can be associated with a point at infinity.The union over all classes of parallels constitute the points of the hyperplane at infinity.…

By a theorem of Hopf and Hirzebruch, closed orientable 4-manifolds always admit a spin c structure.When a manifold carries a spin c structure at all, the set of spin c structures forms an affine space.Moreover, the set of spin c structures has a free transitive action of H 2 (M, Z).…