Euclidean motion

In mathematics, the Euclidean group E(n), also known as ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. Its elements are the isometries associated with the Euclidean distance, and are called Euclidean isometries, Euclidean transformations or Rigid transformations.

Euclidean isometries are classified into direct isometries and indirect isometries, an indirect isometry being an isometry that transforms any object into its mirror image. The direct Euclidean isometries form a group, the special Euclidean group, whose elements are called Euclidean motions, displacements or rigid motions.

These groups are among the oldest and most studied, at least in the cases of dimension 2 and 3 – implicitly, long before the concept of group was invented.

The number of degrees of freedom for E(n) is n(n + 1)/2, which gives 3 in case n = 2, and 6 for n = 3. Of these, n can be attributed to available translational symmetry, and the remaining n(n − 1)/2 to rotational symmetry.

There is a subgroup E⁺(n) of the direct isometries, i.e., isometries preserving orientation, also called rigid motions; they are the moves of a rigid body in n-dimensional space. These include the translations, and the rotations, which together generate E⁺(n). E⁺(n) is also called a special Euclidean group, and denoted SE(n).

The others are the indirect isometries, also called opposite isometries. The subgroup E⁺(n) is of index 2. In other words, the indirect isometries form a single coset of E⁺(n). Given any indirect isometry, for example a given reflection R that reverses orientation, all indirect isometries are given as DR, where D is a direct isometry.

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