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which is a quaternion of unit length (or versor) since ‖ ‖ = + + + = Most importantly, the above equations for composition of rotations are precisely the equations for multiplication of quaternions =. In other words, the group of unit quaternions with multiplication, modulo the negative sign, is isomorphic to the group of rotations with ...
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:
Therefore, a = 0 and b 2 + c 2 + d 2 = 1. 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]
Slerp has a geometric formula independent of quaternions, and independent of the dimension of the space in which the arc is embedded. This formula, a symmetric weighted sum credited to Glenn Davis, is based on the fact that any point on the curve must be a linear combination of the ends.
Like rotation matrices, quaternions must sometimes be renormalized due to rounding errors, to make sure that they correspond to valid rotations. The computational cost of renormalizing a quaternion, however, is much less than for normalizing a 3 × 3 matrix. Quaternions also capture the spinorial character of rotations in three dimensions.
The axes of the original frame are denoted as x, y, z and the axes of the rotated frame as X, Y, Z.The geometrical definition (sometimes referred to as static) begins by defining the line of nodes (N) as the intersection of the planes xy and XY (it can also be defined as the common perpendicular to the axes z and Z and then written as the vector product N = z × Z).
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. This subspace is called an ideal in ring theory. It happens to be the unique maximal ideal of the ring of dual numbers.
In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called.