आइसोमेट्री इसके अंग्रेजी अर्थ का उदाहरण

If h is a translation, then its conjugation by an isometry can be described as applying the isometry to the translation:.

Each isometry is given by an orthogonal matrix A in O(n) and a vector b:.

Two isometry groups are said to be equal up to conjugacy with respect to affine transformations if there is an affine transformation such that all elements of one group are obtained by taking the conjugates by that affine transformation of all elements of the other group.

Note however that the conjugate with respect to an affine transformation of an isometry is in general not an isometry, although volume (in 2D: area) and orientation are preserved.

Consider the 2D isometry point group Dn.

finite cyclic subgroups Cn (n ≥ 1); for every n there is one isometry group, of abstract group type Zn.

finitely generated groups, each isomorphic to one of the form Zm \oplusZ n generated by Cn and m independent rotations with an irrational number of turns, and m, n ≥ 1; for each pair (m, n) there are uncountably many isometry groups, all the same as abstract group; for the pair (1, 1) the group is cyclic.

As topological subgroups of O(2), only the finite isometry groups and SO(2) and O(2) are closed.

The 2D symmetry groups correspond to the isometry groups, except that symmetry according to O(2) and SO(2) can only be distinguished in the generalized symmetry concept applicable for vector fields.

For each of the wallpaper groups p1, p2, p3, p4, p6, the image under p of all isometry groups (i.

The isometry groups of p6m are each mapped to one of the point groups of type D6.

For the other 11 wallpaper groups, each isometry group is mapped to one of the point groups of the types D1, D2, D3, or D4.

Thus, isometry groups of e.

For a given isometry group, the conjugates of a translation in the group by the elements of the group generate a translation group (a lattice)"mdash;that is a subgroup of the isometry group that only depends on the translation we started with, and the point group associated with the isometry group.