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There are 3 × 3 × 3 = 27 possible combinations of three basic rotations but only 3 × 2 × 2 = 12 of them can be used for representing arbitrary 3D rotations as Euler angles. These 12 combinations avoid consecutive rotations around the same axis (such as XXY) which would reduce the degrees of freedom that can be represented.
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.
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]
The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...
For example, a quarter turn around the positive x-axis followed by a quarter turn around the positive y-axis is a different rotation than the one obtained by first rotating around y and then x. The orthogonal group, consisting of all proper and improper rotations, is generated by reflections.
This category deals with topics in physics related to the three-dimensional spherical symmetries of physical objects, including topics concerning rotations in classical mechanics, as well as spin and angular momentum in quantum mechanics, and the representations of the Lie groups SU(2) and SO(3).
The rotation axis is obviously orthogonal to this plane, and passes through the center C of the sphere. Given that for a rigid body any movement that leaves an axis invariant is a rotation, this also proves that any arbitrary composition of rotations is equivalent to a single rotation around a new axis.
Its subgroup of rotations is the dihedral group D n of order 2n, which still has the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes. Note: in 2D, D n includes reflections, which can also be viewed as flipping over flat objects without distinction of frontside and backside; but in 3D, the two operations are ...