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  2. Euclidean group - Wikipedia

    en.wikipedia.org/wiki/Euclidean_group

    The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n), for inhomogeneous special orthogonal group. The Euclidean group E(n) comprises all translations, rotations, and reflections of ; and arbitrary finite combinations of them.

  3. Euclidean space - Wikipedia

    en.wikipedia.org/wiki/Euclidean_space

    The map is a group homomorphism from the Euclidean group onto the group of linear isometries, called the orthogonal group. The kernel of this homomorphism is the translation group, showing that it is a normal subgroup of the Euclidean group.

  4. Isometry group - Wikipedia

    en.wikipedia.org/wiki/Isometry_group

    The isometry group of a two-dimensional sphere is the orthogonal group O(3). [3] The isometry group of the n-dimensional Euclidean space is the Euclidean group E(n). [4] The isometry group of the Poincaré disc model of the hyperbolic plane is the projective special unitary group PSU(1,1).

  5. 3D rotation group - Wikipedia

    en.wikipedia.org/wiki/3D_rotation_group

    The rotation group generalizes quite naturally to n-dimensional Euclidean space, with its standard Euclidean structure. The group of all proper and improper rotations in n dimensions is called the orthogonal group O(n), and the subgroup of proper rotations is called the special orthogonal group SO(n), which is a Lie group of dimension n(n − 1)/2.

  6. List of planar symmetry groups - Wikipedia

    en.wikipedia.org/wiki/List_of_planar_symmetry_groups

    This article summarizes the classes of discrete symmetry groups of the Euclidean plane.The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation.

  7. Point groups in three dimensions - Wikipedia

    en.wikipedia.org/wiki/Point_groups_in_three...

    In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere.It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices.

  8. Orthogonal group - Wikipedia

    en.wikipedia.org/wiki/Orthogonal_group

    Let E(n) be the group of the Euclidean isometries of a Euclidean space S of dimension n. This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are isomorphic. The stabilizer subgroup of a point x ∈ S is the subgroup of the elements g ∈ E(n) such that g(x) = x.

  9. Dihedral group - Wikipedia

    en.wikipedia.org/wiki/Dihedral_group

    In mathematics, a dihedral group is the group of symmetries of a regular polygon, [1] [2] ... is the group of Euclidean plane isometries which keep the origin fixed.