affine space

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  • Info In mathematics, an affine space is a geometric structure that generalizes certain properties of parallel lines in Euclidean space. more...
  • 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 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.
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How affine space gets used