If these conditions do not hold, the formula describes a more general affine transformation of the plane provided that the determinant of A is not zero.

A finite affine plane of order q, with the lines as blocks, is an S ( 2, q, q < sup > 2 </ sup >).

The projective plane of order 2 ( the Fano plane ) is an STS ( 7 ) and the affine plane of order 3 is an STS ( 9 ).

For example, if the affine transformation acts on the plane and if the determinant of A is 1 or − 1 then the transformation is an equi-areal mapping.

To visualise the general affine transformation of the Euclidean plane, take labelled parallelograms ABCD and A ′ B ′ C ′ D ′.

Whatever the choices of points, there is an affine transformation T of the plane taking A to A ′, and each vertex similarly.

The plane may also be viewed as an affine space, whose isomorphisms are combinations of translations and non-singular linear maps.

For any affine transformation of the Gaussian plane, z mapping to a z + b, a ≠ 0, a triangle is transformed but does not change its shape.

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

Some examples are finite sets in ℝ, smooth curves in the plane, and proper affine subspaces in a Euclidean space.

It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface.

In Euclidean plane geometry terms, being a parallelogram is affine since affine transformations always take one parallelogram to another one.

Projective geometry can be modeled by the affine plane ( or affine space ) plus a line ( hyperplane ) " at infinity " and then treating that line ( or hyperplane ) as " ordinary ".

More recently " the affine plane associated to the Lorentzian vector space L < sup > 2 </ sup > " was described by Graciela Birman and Katsumi Nomizu in an article " Trigonometry in Lorentzian geometry ".

An axiomatic treatment of plane affine geometry can be built from the axioms of ordered geometry by the addition of two additional axioms.

Szmielew considers Desarguean as well as Pappian affine plane in the third chapter of From affine to Euclidean geometry.

Finite affine plane of order 2, containing 4 points and 6 lines.

Finite affine plane of order 3, containing 9 points and 12 lines.

The simplest affine plane contains only four points ; it is called the affine plane of order 2.

More generally, a finite affine plane of order n has n < sup > 2 </ sup > points and n < sup > 2 </ sup > + n lines ; each line contains n points, and each point is on n + 1 lines.

The affine plane of order 3 is known as the Hesse configuration.

If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2.

A finite plane of order n is one such that each line has n points ( for an affine plane ), or such that each line has n + 1 points ( for a projective plane ).

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.

In geometry and topology, the line at infinity is a line that is added to the real ( affine ) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane.

In projective geometry, the plane at infinity is a projective plane which is added to the affine 3-space in order to give it closure of incidence properties.

For example, all order automorphisms of a power set S = 2 < sup > R </ sup > are induced by permutations of R. The papers cited above treat only sets S of functions on R < sup > n </ sup > satisfying some condition of convexity and prove that all order automorphisms are induced by linear or affine transformations of R < sup > n </ sup >.

The number n is called the order of the affine plane.

The smallest affine plane ( of order 2 ) is obtained by removing a line ( and the three points on that line ) from the Fano plane.

An affine plane of order n exists if and only if a projective plane of order n exists ( the definitions of order in these cases is not the same ).

Thus, there is no affine plane of order 6 or order 10.

In Drinfeld's approach, quantum groups arise as Hopf algebras depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or affine, when q

If X has dimension at most n and F is a torsion sheaf then these cohomology groups with compact support vanish if q > 2n, and if in addition X is affine of finite type over a separably closed field the cohomology groups vanish for q > n ( for the last statement, see SGA 4, XIV, Cor. 3. 2 ).

* For every finite field F < sub > q </ sub > with q (> 2 ) elements, the group of invertible affine transformations, acting naturally on F < sub > q </ sub > is a Frobenius group.

Then is a translation affine plane of order q < sup > 2 </ sup >.

Using affine transformations multiple different euclidean transformations including translation can be combined by simply multiplying the corresponding matrices.

Choosing different blocks to remove in this way can lead to non-isomorphic affine planes.

If an origin is also chosen, this can be decomposed as an affine transformation that sends, namely

Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by a matrix A and the translation as the addition of a vector, an affine map acting on a vector can be represented as

The above mentioned augmented matrix is called affine transformation matrix, or projective transformation matrix ( as it can also be used to perform Projective transformations ).

The advantage of using homogeneous coordinates is that one can combine any number of affine transformations into one by multiplying the respective matrices.

If there is a fixed point, we can take that as the origin, and the affine transformation reduces to a linear transformation.

The Atbash cipher can be seen as a special case of the affine cipher.

For more complex recognition problems, intelligent character recognition systems are generally used, as artificial neural networks can be made indifferent to both affine and non-linear transformations.

If the affine transformation can be decomposed into isometries and a transformation given by a diagonal matrix, we have directionally differential scaling and the diagonal elements ( the eigenvalues ) are the scale factors in two or three perpendicular directions.

To talk about densities but avoid dealing with measure-theoretic complications it can be simpler to restrict attention to a subset of of the coordinates of such that covariance matrix for this subset is positive definite ; then the other coordinates may be thought of as an affine function of the selected coordinates.

Using the disintegration theorem we can define a restriction of Lebesgue measure to the-dimensional affine subspace of where the Gaussian distribution is supported, i. e..

It can be thought of as a vector space ( or affine space ), a metric space, a topological space, a measure space, or a linear continuum.

* The dimension of a polynomial ring over a field is the number of indeterminates d. These rings correspond to affine spaces in the language of schemes, so this result can be thought of as foundational.

In geometry, a hyperplane of an n-dimensional space V is a " flat " subset of dimension n − 1, or equivalently, of codimension 1 in V ; it may therefore be referred to as an ( n − 1 )- flat of V. The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly ; in all cases however, any hyperplane can be given in coordinates as the solution of a single ( due to the " codimension 1 " constraint ) algebraic equation of degree 1 ( due to the " flat " constraint ).

If V is a vector space, one distinguishes " vector hyperplanes " ( which are subspaces, and therefore must pass through the origin ) and " affine hyperplanes " ( which need not pass through the origin ; they can be obtained by translation of a vector hyperplane ).

The affine group can be viewed as the group of all affine transformations of the affine space underlying the vector space F < sup > n </ sup >.

The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

One variable affine functions can be written as.

When the number of control points exceeds the minimum required to define the appropriate transformation model, iterative algorithms like RANSAC can be used to robustly estimate the parameters of a particular transformation type ( e. g. affine ) for registration of the images.

If X is an affine algebraic set ( irreducible or not ) then the Zariski topology on it is defined simply to be the subspace topology induced by its inclusion into some Equivalently, it can be checked that:

Projective geometry, like affine and Euclidean geometry, can also be developed from the Erlangen program of Felix Klein ; projective geometry is characterized by invariants under transformations of the projective group.