* Affine subspace, linear manifold or linear variety, is a geometric structure that generalizes the affine properties of Euclidean space.

* Affine space, which distinguishes between vectors and points

Affine geometry can be viewed as the geometry of affine space, of a given dimension n, coordinatized over a field K. There is also ( in two dimensions ) a combinatorial generalization of coordinatized affine space, as developed in synthetic finite geometry.

Affine space is usually studied as analytic geometry using coordinates, or equivalently vector spaces.

* Affine space

* Affine group, the group of all invertible affine transformations from any affine space over a field K into itself

* Affine representation, a continuous group homomorphism whose values are automorphisms of an affine space

* Affine space, an abstract structure that generalises the affine-geometric properties of Euclidean space

# REDIRECT Affine space # Affine combinations and affine dependence

Affine space is topologically contractible, so it admits no non-trivial topological vector bundles.

* Affine space

Affine connections may also be used to define ( affine ) geodesics on a manifold, generalizing the straight lines of Euclidean space, although the geometry of those straight lines can be very different from usual Euclidean geometry ; the main differences are encapsulated in the curvature of the connection.

Affine geometry is the geometry of affine space of a given dimension n over a field K. The case where K is the real numbers gives an adequate idea of the content.

# REDIRECT Affine space

# REDIRECT Affine space

# REDIRECT Affine space # Affine combinations and affine dependence

# REDIRECT Affine space # Affine combinations and affine dependence

* Affine space

Affine transformations such as translation, and rotation can be applied on the curve by applying the respective transform on the control points of the curve.

Affine geometry can be developed on the basis of linear algebra.

Affine invariants can also assist calculations.

Affine varieties can be given a natural topology by declaring the closed sets to be precisely the affine algebraic sets.

Affine spaces are subspaces of projective spaces: an affine plane can be obtained from any projective plane by removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane as a closure by adding a line at infinity whose points correspond to equivalence classes of parallel lines.

Affine group varieties are known as linear algebraic groups, since they can be embedded as subgroups of general linear groups.

Affine spaces can be defined in an analogous manner to the construction of affine planes from projective planes.

Affine logic can be embedded into linear logic by rewriting the affine arrow as the linear arrow.

Affine forms can be combined with the standard arithmetic operations or elementary functions, to obtain guaranteed approximations to formulas.

Affine arithmetic can be viewed in matrix form as follows.

Affine arithmetic can be implemented by a global array A and a global vector b, as described above.

Affine holonomy groups are the groups arising as holonomies of torsion-free affine connections ; those which are not Riemannian or pseudo-Riemannian holonomy groups are also known as non-metric holonomy groups.

* Affine combinations are like convex combinations, but the coefficients are not required to be non-negative.

Affine connection is the basis for parallel transport of vectors from one space-time point to another ; Eddington assumed the affine connection to be symmetric in its covariant indices, because it seemed plausible that the result of parallel-transporting one infinitesimal vector along another should produce the same result as transporting the second along the first.

Affine arithmetic is meant to be an improvement on interval arithmetic ( IA ), and is similar to generalized interval arithmetic, first-order Taylor arithmetic, the center-slope model, and ellipsoid calculus — in the sense that it is an automatic method to derive first-order guaranteed approximations to general formulas.

* Affine vector field

Affine vector fields ( affines ) satisfy ( equivalently, ) and hence every affine is a projective.

* Affine vector field

Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction.

Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality.

Affine varieties of non-zero dimension are never proper.

Affine gap penalties use a gap opening penalty, o, and a gap extension penalty, e. A gap of length l is then given a penalty o + ( l-1 ) e. So that gaps are discouraged, o is almost always negative.

Affine Lie algebras play an important role in string theory and conformal field theory due to the way they are constructed: starting from a simple Lie algebra, one considers the loop algebra,, formed by the-valued functions on a circle ( interpreted as the closed string ) with pointwise commutator.

Affine transformations are applied to these polytopes, producing a description of a new execution order.