Search results
Results from the WOW.Com Content Network
A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set. In pseudo-Euclidean space the metric is replaced with an isotropic quadratic form ; transformations preserving this form are sometimes called "isometries", and the collection of them is ...
A collection of isometries typically form a group, the isometry group. When the group is a continuous group, the infinitesimal generators of the group are the Killing vector fields. The Myers–Steenrod theorem states that every isometry between two connected Riemannian manifolds is smooth (differentiable).
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.
The Euclidean group is a subgroup of the group of affine transformations. It has as subgroups the translational group T( n ), and the orthogonal group O( n ). Any element of E( n ) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way: x ↦ A ( x + b ) {\displaystyle x\mapsto A(x+b)} where ...
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)—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. This is because the conjugate of the translation ...
An isometry of the Euclidean plane is a distance-preserving transformation of the plane. ... This set forms a group under composition, the group of rigid motions, ...
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d).
The Poincaré group, named after Henri Poincaré (1905), [1] was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. [ 2 ] [ 3 ] It is a ten-dimensional non-abelian Lie group that is of importance as a model in our understanding of the most basic fundamentals of physics .