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Each rotation is represented by two unit quaternions of opposite sign, and, as in the space of rotations in three dimensions, the quaternion product of two unit quaternions will yield a unit quaternion. Also, the space of unit quaternions is "flat" in any infinitesimal neighborhood of a given unit quaternion.
There are two representations of quaternions. This article uses the more popular Hamilton. A quaternion has 4 real values: q w (the real part or the scalar part) and q x q y q z (the imaginary part). Defining the norm of the quaternion as follows: ‖ ‖ = + + +
Two reflections make a rotation by an angle twice the angle between the two reflection planes, so ′ ′ = corresponds to a rotation of 180° in the plane containing σ 1 and σ 2. This is very similar to the corresponding quaternion formula,
The number of Euler angles needed to represent the group SO(n) is n(n − 1)/2, equal to the number of planes containing two distinct coordinate axes in n-dimensional Euclidean space. In SO(4) a rotation matrix is defined by two unit quaternions, and therefore has six degrees of freedom, three from each quaternion.
Quaternions also capture the spinorial character of rotations in three dimensions. For a three-dimensional object connected to its (fixed) surroundings by slack strings or bands, the strings or bands can be untangled after two complete turns about some fixed axis from an initial untangled state. Algebraically, the quaternion describing such a ...
where v is the rotation vector treated as a quaternion. A single multiplication by a versor, either left or right, is itself a rotation, but in four dimensions. Any four-dimensional rotation about the origin can be represented with two quaternion multiplications: one left and one right, by two different unit quaternions.
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The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations. The map from S 3 onto SO(3) that identifies antipodal points of S 3 is a surjective homomorphism of Lie groups, with kernel {±1}. Topologically, this map is a two-to-one covering map. (See the plate trick.)