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

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

#( Affine axiom of parallelism ) Given a point A and a line r, not through A, there is at most one line through A which does not meet r.

Affine invariants can also assist calculations.

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 can also be viewed as a vector space whose operations are limited to those linear combinations whose coefficients sum to one, for example 2x − y, x − y + z, ( x + y + z )/ 3, ix +( 1-i ) y, etc.

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.

* Emil Artin ( 1957 ) Geometric Algebra, chapter 2: " Affine and projective geometry ", Interscience Publishers.

Ashkinuse & Isaak Yaglom ( 1962 ) Ideas and Methods of Affine and Projective Geometry ( in Russian ), Ministry of Education, Moscow.

* M. K. Bennett ( 1995 ) Affine and Projective Geometry, John Wiley & Sons ISBN 0-471-11315-8.

* H. S. M. Coxeter ( 1955 ) " The Affine Plane ", Scripta Mathematica 21: 5 – 14, a lecture delivered before the Forum of the Society of Friends of Scripta Mathematica on Monday, April 26, 1954.

* Bruce E. Meserve ( 1955 ) Fundamental Concepts of Geometry, Chapter 5 Affine Geometry ,, pp 150 – 84, Addison-Wesley.

* Wanda Szmielew ( 1984 ) From Affine to Euclidean Geometry: an axiomatic approach, D. Reidel, ISBN 90-277-1243-3.

* Oswald Veblen ( 1918 ) Projective Geometry, volume 2, chapter 3: Affine group in the plane, pp 70 to 118, Ginn & Company.

* Peter Cameron Projective and Affine Geometries from University of London.

* Jean H. Gallier ( 2001 ) Geometric Methods and Applications for Computer Science and Engineering, Chapter 2: Basics of Affine Geometry, Springer Texts in Applied Mathematics # 38, chapter online from University of Pennsylvania ( PDF ).

When one wants to work over a base ring R ( commutative ), there is the group scheme concept: that is, a group object in the category of schemes over R. Affine group scheme is the concept dual to a type of Hopf algebra.

Affine and projective planes of order n exist whenever n is a prime power ( a prime number raised to a positive integer exponent ), by using affine and projective planes over the finite field with n = p < sup > k </ sup > elements.

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

The Caesar cipher is the Affine cipher when since the encrypting function simply reduces to a linear shift.

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

Affine functions ( the function is an example ) do not scale multiplicatively.

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

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.

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

Affine arithmetic ( AA ) is a model for self-validated numerical analysis.

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 arithmetic is potentially useful in every numeric problem where one needs guaranteed enclosures to smooth functions, such as solving systems of non-linear equations, analyzing dynamical systems, integrating functions differential equations, etc.

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

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

Affine varieties of non-zero dimension are never proper.

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 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 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.

A common extension to standard linear gap costs, is the usage of two different gap penalties for opening a gap and for extending a gap.

Days later, Bayern won the Champions League for the fourth time after a 25-year gap, defeating Valencia CF on penalties.

Note that and depend upon the substitution matrix, gap penalties, and sequence composition ( the letter frequencies ).

Note that the computation of the score and its corresponding E score is involved with the adequate gap penalties.

Gap penalties contribute to the overall score of alignments, and therefore, the size of the gap penalty relative to the entries in the similarity matrix affects the alignment that is finally selected.

Constant gap penalties are the simplest type of gap penalty.

Linear gap penalties have only one parameter, d, which is a penalty per unit length of gap.

By the late nineteen-thirties the widespread adoption of all-metal stressed skin construction of aircraft meant, at least in theory, that the aerodynamic penalties that had limited the performance of pushers ( and indeed any unconventional layout ), were largely negated ; however any improvement that boosts pusher performance also boosts the performance of conventional aircraft and they remained a rarity in operational service – so the gap was narrowed but not closed entirely.

The expectation score is defined as the average score that the scoring system ( substitution matrix and gap penalties ) would yield for a random sequence.