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.…

In affine space, the union of a line and a point not on the line is not equidimensional.In general, if two closed subschemes of some scheme, neither containing the other, have unequal dimensions, then their union is not equidimensional.…

Thus the normal affine space is the plane of equation x=a.

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.…

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.…

It can also be studied as synthetic geometry by writing down axioms, though this approach is much less common.There are several different systems of axioms for affine space.…

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.…

A 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.A closed half-space is the union of an open half-space and the hyperplane that defines it.…

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).…

This motivates the subscripts and for identifying the null basis vectors.The choice of the origin is arbitrary: any other point may be chosen, as the representation is of an affine space.The origin merely represents a reference point, and is algebraically equivalent to any other point.…

The distinction says that there is no canonical choice of where the origin should go in an affine -space, because it can be translated anywhere.

They divided the 219 affine space groups into reducible and irreducible groups.The reducible groups fall into 17 classes corresponding to the 17 wallpaper groups, and the remaining 35 irreducible groups are the same as the cubic groups and are classified separately.…

A bounded domain is an open connected bounded subset of a complex affine space.It is called homogeneous if its group of automorphisms acts transitively, and is called symmetric if for every point there is an automorphism acting as -1 on the tangent space.…