This HTML5 document contains 95 embedded RDF statements represented using HTML+Microdata notation.

The embedded RDF content will be recognized by any processor of HTML5 Microdata.

Namespace Prefixes

PrefixIRI
dcthttp://purl.org/dc/terms/
yago-reshttp://yago-knowledge.org/resource/
dbohttp://dbpedia.org/ontology/
foafhttp://xmlns.com/foaf/0.1/
n10http://dbpedia.org/resource/File:
dbthttp://dbpedia.org/resource/Template:
rdfshttp://www.w3.org/2000/01/rdf-schema#
freebasehttp://rdf.freebase.com/ns/
n12http://commons.wikimedia.org/wiki/Special:FilePath/
rdfhttp://www.w3.org/1999/02/22-rdf-syntax-ns#
owlhttp://www.w3.org/2002/07/owl#
n9http://en.wikipedia.org/wiki/
dbphttp://dbpedia.org/property/
provhttp://www.w3.org/ns/prov#
dbchttp://dbpedia.org/resource/Category:
xsdhhttp://www.w3.org/2001/XMLSchema#
dbrhttp://dbpedia.org/resource/

Statements

Subject Item
dbr:Euler's_rotation_theorem
rdf:type
owl:Thing
rdfs:label
Euler's rotation theorem
rdfs:comment
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group.
rdfs:seeAlso
dbr:4-dimensional_Euclidean_space dbr:Rotation
owl:sameAs
freebase:m.03jpn0 yago-res:Euler's_rotation_theorem
dbp:wikiPageUsesTemplate
dbt:Efn dbt:Leonhard_Euler dbt:Math dbt:Sup dbt:Mvar dbt:See_also dbt:Cite_journal dbt:Notelist dbt:Refimprove dbt:Main dbt:Clear dbt:= dbt:Sfrac dbt:Citizendium dbt:Short_description
dct:subject
dbc:Rotation_in_three_dimensions dbc:Euclidean_symmetries dbc:Theorems_in_geometry dbc:Leonhard_Euler
dbo:thumbnail
n12:Euler_AxisAngle.png?width=300
foaf:depiction
n12:Pure_screw.svg n12:Euler_Rotation_1.jpg n12:Euler_Rotation_3.jpg n12:Euler_Rotation_2.jpg n12:Eulerrotation.svg n12:Euler_AxisAngle.png
prov:wasDerivedFrom
n9:Euler's_rotation_theorem?oldid=1073598724&ns=0
dbo:wikiPageID
865138
dbo:wikiPageLength
29195
dbo:wikiPageRevisionID
1073598724
dbo:wikiPageWikiLink
dbr:Matrix_exponential dbr:Euler–Rodrigues_formula dbr:Kinematics dbr:Improper_rotation dbr:Chasles'_theorem_(kinematics) dbr:William_Rowan_Hamilton n10:Euler_AxisAngle.png dbr:Plane_of_rotation n10:Euler_Rotation_2.JPG dbr:Equivalence_relation dbr:Cartesian_coordinate_system n10:Eulerrotation.svg dbr:Vector_(mathematics_and_physics) dbr:Trace_(linear_algebra) dbr:Rigid_body dbr:Screw_theory dbr:Rotation_around_a_fixed_axis dbr:Normal_matrix dbr:Eigenvalues_and_eigenvectors dbr:Rotation_formalisms_in_three_dimensions dbr:Computer_graphics n10:Euler_Rotation_1.JPG n10:Euler_Rotation_3.JPG dbr:Rotation_matrix dbr:Fixed_point_(mathematics) dbr:Three-dimensional_space dbr:Change_of_basis dbr:Rotation_(mathematics) dbr:Group_(mathematics) dbr:Unit_vector dbr:3D_rotation_group dbr:Complex_number dbr:Kernel_(linear_algebra) n10:Pure_screw.svg dbr:Screw_axis dbc:Euclidean_symmetries dbr:Axis–angle_representation dbr:Spherical_trigonometry dbr:Leonhard_Euler dbr:Direction_cosine dbr:Round-off_error dbr:Euler_angles dbc:Rotation_in_three_dimensions dbr:Angular_velocity dbc:Leonhard_Euler dbr:Characteristic_polynomial dbr:Determinant dbr:Geometry dbr:Projection_(linear_algebra) dbr:Spherical_geometry dbr:Lie_algebra dbr:Orthogonal_matrix dbr:Complex_conjugate dbr:Instant_centre_of_rotation dbc:Theorems_in_geometry dbr:Quaternion
dbo:abstract
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group. The theorem is named after Leonhard Euler, who proved it in 1775 by means of spherical geometry. The axis of rotation is known as an Euler axis, typically represented by a unit vector ê. Its product by the rotation angle is known as an axis-angle vector. The extension of the theorem to kinematics yields the concept of instant axis of rotation, a line of fixed points. In linear algebra terms, the theorem states that, in 3D space, any two Cartesian coordinate systems with a common origin are related by a rotation about some fixed axis. This also means that the product of two rotation matrices is again a rotation matrix and that for a non-identity rotation matrix one eigenvalue is 1 and the other two are both complex, or both equal to −1. The eigenvector corresponding to this eigenvalue is the axis of rotation connecting the two systems.
foaf:isPrimaryTopicOf
n9:Euler's_rotation_theorem