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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). The isometry group of the Poincaré half-plane model of ...
An object having symmetry group D n, D nh, or D nd has rotation group D n. An object having a polyhedral symmetry (T, T d, T h, O, O h, I or I h) has as its rotation group the corresponding one without a subscript: T, O or I. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral. In other words ...
A path isometry or arcwise isometry is a map which preserves the lengths of curves; such a map is not necessarily an isometry in the distance preserving sense, and it need not necessarily be bijective, or even injective. [5] [6] This term is often abridged to simply isometry, so one should take care to determine from context which type is intended.
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 ...
A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space. For an object, any unique centre and, more generally, any point with unique properties with respect to the object is a fixed point of its ...
Any object will keep the same shape and size after a proper rigid transformation. All rigid transformations are examples of affine transformations. The set of all (proper and improper) rigid transformations is a mathematical group called the Euclidean group, denoted E(n) for n-dimensional Euclidean spaces.
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 ...
It is a semidirect product of R n with a cyclic group of order 2, the latter acting on R n by negation. It is precisely the subgroup of the Euclidean group that fixes the line at infinity pointwise. In the case n = 1, the point reflection group is the full isometry group of the line.