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In other words: A quaternion squares to −1 if and only if it is a vector quaternion with norm 1. By definition, the set of all such vectors forms the unit sphere. Only negative real quaternions have infinitely many square roots. All others have just two (or one in the case of 0). [citation needed] [d]
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]
Quaternion variable theory differs in some respects from complex variable theory. For example: The complex conjugate mapping of the complex plane is a central tool but requires the introduction of a non-arithmetic, non-analytic operation.
The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere S 3 and can be understood as the group of versors (quaternions with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations.
The product of a quaternion with its conjugate is its common norm. [63] The operation of taking the common norm of a quaternion is represented with the letter N. By definition the common norm is the product of a quaternion with its conjugate. It can be proven [64] [65] that common norm is equal to the square of the tensor of a quaternion ...
The full automorphism group of Q 8 is isomorphic to S 4, the symmetric group on four letters (see Matrix representations below), and the outer automorphism group of Q 8 is thus S 4 /V, which is isomorphic to S 3. The quaternion group Q 8 has five conjugacy classes, {}, {¯}, {, ¯}, {, ¯}, {, ¯}, and so five irreducible representations over ...
In mathematics, a rotor in the geometric algebra of a vector space V is the same thing as an element of the spin group Spin(V).We define this group below. Let V be a vector space equipped with a positive definite quadratic form q, and let Cl(V) be the geometric algebra associated to V.
If p + ε q is a dual quaternion, and p is not zero, then the inverse dual quaternion is given by p −1 (1 − ε q p −1 ). Thus the elements of the subspace { ε q : q ∈ H } do not have inverses.
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