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Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 [1] [2] and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by H (for Hamilton ), or in blackboard bold by H . {\displaystyle \mathbb {H} .}
In mathematics, quaternions are a non-commutative number system that extends the complex numbers.Quaternions and their applications to rotations were first described in print by Olinde Rodrigues in all but name in 1840, [1] but independently discovered by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space.
William Rowan Hamilton invented quaternions, a mathematical entity in 1843. This article describes Hamilton's original treatment of quaternions, using his notation and terms. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. Mathematically, quaternions discussed differ from the ...
It can also be realized as the subgroup of unit quaternions generated by [10] = / and =. The generalized quaternion groups have the property that every abelian subgroup is cyclic. [ 11 ] It can be shown that a finite p -group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined ...
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]
It is famous for being the location where Sir William Rowan Hamilton first wrote down the fundamental formula for quaternions on 16 October 1843, which is to this day commemorated by a stone plaque on the northwest corner of the underside of the bridge.
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:
Hamilton's primary exposition on biquaternions came in 1853 in his Lectures on Quaternions. The editions of Elements of Quaternions , in 1866 by William Edwin Hamilton (son of Rowan), and in 1899, 1901 by Charles Jasper Joly , reduced the biquaternion coverage in favour of the real quaternions.