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The rotation is completely specified by specifying the axis planes and the angles of rotation about them. Without loss of generality, these axis planes may be chosen to be the uz - and xy-planes of a Cartesian coordinate system, allowing a simpler visualization of the rotation. In 4D space, the Hopf angles {ξ 1, η, ξ 2} parameterize the 3 ...
For example, in 2-space n = 2, a rotation by angle θ has eigenvalues λ = e iθ and λ = e −iθ, so there is no axis of rotation except when θ = 0, the case of the null rotation. In 3-space n = 3, the axis of a non-null proper rotation is always a unique line, and a rotation around this axis by angle θ has eigenvalues λ = 1, e iθ, e −iθ.
Einstein's concept of spacetime has a Minkowski structure based on a non-Euclidean geometry with three spatial dimensions and one temporal dimension, rather than the four symmetric spatial dimensions of Schläfli's Euclidean 4D space. Single locations in Euclidean 4D space can be given as vectors or 4-tuples, i.e., as ordered lists of numbers ...
3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]
If a rotation of Minkowski space is in a space-like plane, then this rotation is the same as a spatial rotation in Euclidean space. By contrast, a rotation in a plane spanned by a space-like dimension and a time-like dimension is a hyperbolic rotation, and if this plane contains the time axis of the reference frame, is called a "Lorentz boost ...
Rotation formalisms are focused on proper (orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to.Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions.
1951, A. C. Hurley, Finite rotation groups and crystal classes in four dimensions, Proceedings of the Cambridge Philosophical Society, vol. 47, issue 04, p. 650 [1] 1962 A. L. MacKay Bravais Lattices in Four-dimensional Space [2] 1964 Patrick du Val, Homographies, quaternions and rotations, quaternion-based 4D point groups
Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates (i.e. it can't be combined with other transformations while preserving commutativity and other properties), it becomes, in a 3-D or 4-D projective space described by homogeneous coordinates, a simple linear transformation (a ...