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There are at least two ways of representing quaternions as matrices in such a way that quaternion addition and multiplication correspond to matrix addition and matrix multiplication. One is to use 2 × 2 complex matrices, and the other is to use 4 × 4 real matrices. In each case, the representation given is one of a family of linearly related ...
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 most external matrix rotates the other two, leaving the second rotation matrix over the line of nodes, and the third one in a frame comoving with the body. 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 ...
In mathematics, a quaternion algebra over a field F is a central simple algebra A over F [1] [2] that has dimension 4 over F.Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, is isomorphic to the 2 × 2 matrix algebra over K.
The quaternion group has the unusual property of being Hamiltonian: Q 8 is non-abelian, but every subgroup is normal. [4] Every Hamiltonian group contains a copy of Q 8. [5] The quaternion group Q 8 and the dihedral group D 4 are the two smallest examples of a nilpotent non-abelian group.
A direct formula for the conversion from a quaternion to Euler angles in any of the 12 possible sequences exists. [2] For the rest of this section, the formula for the sequence Body 3-2-1 will be shown. If the quaternion is properly normalized, the Euler angles can be obtained from the quaternions via the relations:
The Lie algebra of SO(3) is denoted by () and consists of all skew-symmetric 3 × 3 matrices. [7] This may be seen by differentiating the orthogonality condition , A T A = I , A ∈ SO(3) . [ nb 2 ] The Lie bracket of two elements of s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} is, as for the Lie algebra of every matrix group, given by the ...
By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra so(3) to its Lie group SO(3).