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Euclidean group - Subgroup structure matrix and vector representation | A Wisdom Archive on Euclidean group - Subgroup structure matrix and vector representation | | Euclidean group - Subgroup structure matrix and vector representation A selection of articles related to Euclidean group - Subgroup structure matrix and vector representation | |
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More material related to Euclidean Group can be found here:
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Euclidean group, Euclidean group - Commuting isometries, Euclidean group - Conjugacy classes, Euclidean group - Overview of isometries in up to three dimensions, Euclidean group - Relation to the affine group, Euclidean group - Rigid body motions, Euclidean group - Subgroup structure matrix and vector representation, Euclidean group - Subgroups, fixed points of isometry groups in Euclidean space, Euclidean plane isometry, Poincaré group
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ARTICLES RELATED TO Euclidean group - Subgroup structure matrix and vector representation | |
| | |  Euclidean group - Subgroup structure matrix and vector representation: Encyclopedia II - Euclidean group - SubgroupsTypes of subgroups of E(n):
Finite groups. They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal (with respect to inclusion) among the finite groups: Oh and Ih. The groups Ih are even maximal among the groups including the next category.
Countably infinite groups without arbitrarily small translations, rotations, or combinations, i.e., for every point the set of images under the isometries is topologically ...
See also:Euclidean group, Euclidean group - Subgroup structure matrix and vector representation, Euclidean group - Subgroups, Euclidean group - Relation to the affine group, Euclidean group - Rigid body motions, Euclidean group - Overview of isometries in up to three dimensions, Euclidean group - Commuting isometries, Euclidean group - Conjugacy classes Read more here: » Euclidean group: Encyclopedia II - Euclidean group - Subgroups |
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| | | | Euclidean group - Subgroup structure matrix and vector representation: Encyclopedia II - Euclidean group - Overview of isometries in up to three dimensions E(1), E(2), and E(3) can be categorized as follows, with degrees of freedom:
E(1) - 1:
E+(1):
identity - 0
translation - 1
those not preserving orientation:
reflection in a point - 1
E(2) - 3:
E+(2):
identity - 0
translation - 2
rotation about a point - 3
those not preserving orientation:
reflection in a line - 2
reflection in a line ...
See also:Euclidean group, Euclidean group - Subgroup structure matrix and vector representation, Euclidean group - Subgroups, Euclidean group - Relation to the affine group, Euclidean group - Rigid body motions, Euclidean group - Overview of isometries in up to three dimensions, Euclidean group - Commuting isometries, Euclidean group - Conjugacy classes Read more here: » Euclidean group: Encyclopedia II - Euclidean group - Overview of isometries in up to three dimensions |
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| | | | Euclidean group - Subgroup structure matrix and vector representation: Encyclopedia II - Euclidean group - Relation to the affine group The Euclidean group E(n) is a subgroup of the affine group for n dimensions, and in such a way as to respect the semidirect product structure of both groups. Instead of by a pair (A, b), Euclidean group elements can also be represented as square matrices of size n + 1, as explained for the affine group.
In the terms of the Erlangen programme, Euclidean geometry is therefore a specialisation of affine geometry. All affine theorems apply; the extra fa ...
See also:Euclidean group, Euclidean group - Subgroup structure matrix and vector representation, Euclidean group - Subgroups, Euclidean group - Relation to the affine group, Euclidean group - Rigid body motions, Euclidean group - Overview of isometries in up to three dimensions, Euclidean group - Commuting isometries, Euclidean group - Conjugacy classes Read more here: » Euclidean group: Encyclopedia II - Euclidean group - Relation to the affine group |
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