Search results
Results from the WOW.Com Content Network
In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line.
Another way to express the definition is that an affine space is a principal homogeneous space for the action of the additive group of a vector space. Homogeneous spaces are, by definition, endowed with a transitive group action, and for a principal homogeneous space, such a transitive action is, by definition, free.
For example, if the affine transformation acts on the plane and if the determinant of is 1 or −1 then the transformation is an equiareal mapping. Such transformations form a subgroup called the equi-affine group. [13] A transformation that is both equi-affine and a similarity is an isometry of the plane taken with Euclidean distance
The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric group S̃ 3. The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher-dimensional objects.
The affine group acts transitively on the points of an affine space, and the subgroup V of the affine group (that is, a vector space) has transitive and free (that is, regular) action on these points; [14] indeed this can be used to give a definition of an affine space.
Among the examples above the additive, multiplicative groups and the general and special linear groups are affine. Using the action of an affine algebraic group on its coordinate ring it can be shown that every affine algebraic group is a linear (or matrix group), meaning that it is isomorphic to an algebraic subgroup of the general linear group.
The affine group Aff(n, F) is an extension of GL(n, F) by the group of translations in F n. It can be written as a semidirect product: Aff(n, F) = GL(n, F) ⋉ F n. where GL(n, F) acts on F n in the natural manner. The affine group can be viewed as the group of all affine transformations of the affine space underlying the vector space F n.
In this viewpoint, an affine transformation is a projective transformation that does not permute finite points with points at infinity, and affine transformation geometry is the study of geometrical properties through the action of the group of affine transformations.