Matches in DBpedia 2015-10 for { <http://dbpedia.org/resource/Rotation_group_SO(3)> ?p ?o }
- Rotation_group_SO(3) abstract "In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e. handedness of space). Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties (along with the associative property, which rotations obey), the set of all rotations is a group under composition. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smooth; so it is in fact a Lie group. It is compact and has dimension 3. Rotations are linear transformations of R3 and can therefore be represented by matrices once a basis of R3 has been chosen. Specifically, if we choose an orthonormal basis of R3, every rotation is described by an orthogonal 3x3 matrix (i.e. a 3x3 matrix with real entries which, when multiplied by its transpose, results in the identity matrix) with determinant 1. The group SO(3) can therefore be identified with the group of these matrices under matrix multiplication. These matrices are known as "special orthogonal matrices", explaining the notation SO(3).The group SO(3) is used to describe the possible rotational symmetries of an object, as well as the possible orientations of an object in space. Its representations are important in physics, where they give rise to the elementary particles of integer spin.".
- Rotation_group_SO(3) wikiPageExternalLink 2000-201.pdf.
- Rotation_group_SO(3) wikiPageID "173965".
- Rotation_group_SO(3) wikiPageLength "60399".
- Rotation_group_SO(3) wikiPageOutDegree "206".
- Rotation_group_SO(3) wikiPageRevisionID "675706397".
- Rotation_group_SO(3) wikiPageWikiLink 3-sphere.
- Rotation_group_SO(3) wikiPageWikiLink Absolute_value.
- Rotation_group_SO(3) wikiPageWikiLink Adjoint_representation.
- Rotation_group_SO(3) wikiPageWikiLink Angle.
- Rotation_group_SO(3) wikiPageWikiLink Angle_of_rotation.
- Rotation_group_SO(3) wikiPageWikiLink Angular_momentum.
- Rotation_group_SO(3) wikiPageWikiLink Antipodal_point.
- Rotation_group_SO(3) wikiPageWikiLink Associative_property.
- Rotation_group_SO(3) wikiPageWikiLink Axis–angle_representation.
- Rotation_group_SO(3) wikiPageWikiLink BCH_formula.
- Rotation_group_SO(3) wikiPageWikiLink Baker–Campbell–Hausdorff_formula.
- Rotation_group_SO(3) wikiPageWikiLink Basis_(linear_algebra).
- Rotation_group_SO(3) wikiPageWikiLink Basis_of_a_vector_space.
- Rotation_group_SO(3) wikiPageWikiLink Bilinear_transform.
- Rotation_group_SO(3) wikiPageWikiLink Block_diagonal_matrix.
- Rotation_group_SO(3) wikiPageWikiLink Block_matrix.
- Rotation_group_SO(3) wikiPageWikiLink Boson.
- Rotation_group_SO(3) wikiPageWikiLink Cartan–Dieudonné_theorem.
- Rotation_group_SO(3) wikiPageWikiLink Casimir_element.
- Rotation_group_SO(3) wikiPageWikiLink Category:Euclidean_solid_geometry.
- Rotation_group_SO(3) wikiPageWikiLink Category:Lie_groups.
- Rotation_group_SO(3) wikiPageWikiLink Category:Rotation_in_three_dimensions.
- Rotation_group_SO(3) wikiPageWikiLink Category:Rotational_symmetry.
- Rotation_group_SO(3) wikiPageWikiLink Charts_on_SO(3).
- Rotation_group_SO(3) wikiPageWikiLink Chirality_(mathematics).
- Rotation_group_SO(3) wikiPageWikiLink Classical_group.
- Rotation_group_SO(3) wikiPageWikiLink Classical_mechanics.
- Rotation_group_SO(3) wikiPageWikiLink Clebsch–Gordan_coefficients.
- Rotation_group_SO(3) wikiPageWikiLink Clockwise.
- Rotation_group_SO(3) wikiPageWikiLink Clockwise_and_counterclockwise.
- Rotation_group_SO(3) wikiPageWikiLink Closed_subgroup_theorem.
- Rotation_group_SO(3) wikiPageWikiLink Commutation_relation.
- Rotation_group_SO(3) wikiPageWikiLink Commutator.
- Rotation_group_SO(3) wikiPageWikiLink Compact_space.
- Rotation_group_SO(3) wikiPageWikiLink Computer_graphics.
- Rotation_group_SO(3) wikiPageWikiLink Connectedness.
- Rotation_group_SO(3) wikiPageWikiLink Coordinate_rotation.
- Rotation_group_SO(3) wikiPageWikiLink Counterclockwise.
- Rotation_group_SO(3) wikiPageWikiLink Covering_group.
- Rotation_group_SO(3) wikiPageWikiLink Covering_map.
- Rotation_group_SO(3) wikiPageWikiLink Covering_space.
- Rotation_group_SO(3) wikiPageWikiLink Cross_product.
- Rotation_group_SO(3) wikiPageWikiLink Cyclic_group.
- Rotation_group_SO(3) wikiPageWikiLink Determinant.
- Rotation_group_SO(3) wikiPageWikiLink Diffeomorphism.
- Rotation_group_SO(3) wikiPageWikiLink Differentiable_manifold.
- Rotation_group_SO(3) wikiPageWikiLink Dimension_(vector_space).
- Rotation_group_SO(3) wikiPageWikiLink Dot_product.
- Rotation_group_SO(3) wikiPageWikiLink Elementary_particle.
- Rotation_group_SO(3) wikiPageWikiLink Euclidean_distance.
- Rotation_group_SO(3) wikiPageWikiLink Euclidean_group.
- Rotation_group_SO(3) wikiPageWikiLink Euclidean_space.
- Rotation_group_SO(3) wikiPageWikiLink Euler_angle.
- Rotation_group_SO(3) wikiPageWikiLink Euler_angles.
- Rotation_group_SO(3) wikiPageWikiLink Eulers_rotation_theorem.
- Rotation_group_SO(3) wikiPageWikiLink Faithful_representation.
- Rotation_group_SO(3) wikiPageWikiLink Fermion.
- Rotation_group_SO(3) wikiPageWikiLink Finite-dimensional.
- Rotation_group_SO(3) wikiPageWikiLink Function_composition.
- Rotation_group_SO(3) wikiPageWikiLink Fundamental_group.
- Rotation_group_SO(3) wikiPageWikiLink General_linear_group.
- Rotation_group_SO(3) wikiPageWikiLink Geodesic.
- Rotation_group_SO(3) wikiPageWikiLink Geometric_algebra.
- Rotation_group_SO(3) wikiPageWikiLink Geometry.
- Rotation_group_SO(3) wikiPageWikiLink Group_(mathematics).
- Rotation_group_SO(3) wikiPageWikiLink Group_contraction.
- Rotation_group_SO(3) wikiPageWikiLink Group_homomorphism.
- Rotation_group_SO(3) wikiPageWikiLink Group_representation.
- Rotation_group_SO(3) wikiPageWikiLink Hairy_ball_theorem.
- Rotation_group_SO(3) wikiPageWikiLink Hermitian_matrix.
- Rotation_group_SO(3) wikiPageWikiLink Hilbert–Schmidt_inner_product.
- Rotation_group_SO(3) wikiPageWikiLink Hilbert–Schmidt_operator.
- Rotation_group_SO(3) wikiPageWikiLink Homeomorphic.
- Rotation_group_SO(3) wikiPageWikiLink Homeomorphism.
- Rotation_group_SO(3) wikiPageWikiLink Homomorphism.
- Rotation_group_SO(3) wikiPageWikiLink Ideal_(Lie_algebra).
- Rotation_group_SO(3) wikiPageWikiLink Identity_function.
- Rotation_group_SO(3) wikiPageWikiLink Identity_map.
- Rotation_group_SO(3) wikiPageWikiLink Identity_matrix.
- Rotation_group_SO(3) wikiPageWikiLink Infinitesimal_rotation.
- Rotation_group_SO(3) wikiPageWikiLink Inverse_function.
- Rotation_group_SO(3) wikiPageWikiLink Invertible_matrix.
- Rotation_group_SO(3) wikiPageWikiLink Isometry.
- Rotation_group_SO(3) wikiPageWikiLink Isomorphic.
- Rotation_group_SO(3) wikiPageWikiLink Isomorphism.
- Rotation_group_SO(3) wikiPageWikiLink Kernel_(algebra).
- Rotation_group_SO(3) wikiPageWikiLink Kernel_(linear_algebra).
- Rotation_group_SO(3) wikiPageWikiLink Kronecker_delta.
- Rotation_group_SO(3) wikiPageWikiLink Kronecker_product.
- Rotation_group_SO(3) wikiPageWikiLink Ladder_operator.
- Rotation_group_SO(3) wikiPageWikiLink Levi-Civita_symbol.
- Rotation_group_SO(3) wikiPageWikiLink Lie_algebra.
- Rotation_group_SO(3) wikiPageWikiLink Lie_algebra_representation.
- Rotation_group_SO(3) wikiPageWikiLink Lie_bracket.