## affine spaces

22 examples (0.01 sec)
• In an affine space, there is no distinguished point that serves as an origin.
• In particular, a vector space is an affine space over itself, by the map.
• More generally, a half-space is either of the two parts into which a hyperplane divides an affine space.
• In an affine space, there are instead displacement vectors between two points of the space.
• In affine space, the union of a line and a point not on the line is not equidimensional.
• Thus the normal affine space is the plane of equation x=a.
• The confusion therefore arises because an affine space with a marked point can be identified with its tangent space at that point.
• Every linear space is also an affine space.
• Informally, an affine space is a vector space without a fixed choice of origin.
• More precisely, an affine space is a set with a free transitive vector space action.
• Then the set complement is called an affine space.
• There are several different systems of axioms for affine space.
• For any isometry group in Euclidean space the set of fixed points is either empty or an affine space.
• An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space.
• Similarly, starting from an affine space A, every class of parallel lines can be associated with a point at infinity.
• When a manifold carries a spin c structure at all, the set of spin c structures forms an affine space.
• The choice of the origin is arbitrary: any other point may be chosen, as the representation is of an affine space.
• 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.
• A bounded domain is an open connected bounded subset of a complex affine space.