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Therefore, nonzero, non-scalar quaternions, or positive scalar quaternions, have exactly two roots, while 0 has exactly one root (0), and negative scalar quaternions have infinitely many roots, which are the vector quaternions located on {} (), i.e., where the scalar part is zero and the vector part is located on the 2-sphere with radius .
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]
These two scalars (negative and positive unity) can be thought of as scalar quaternions. These two scalars are special limiting cases, corresponding to versors with angles of either zero or π. These two scalars are special limiting cases, corresponding to versors with angles of either zero or π.
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:
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.
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.
Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called. As with complex and real analysis , it is possible to study the concepts of analyticity , holomorphy , harmonicity and conformality in the context of quaternions.
The quaternion formulation of the composition of two rotations R B and R A also yields directly the rotation axis and angle of the composite rotation R C = R B R A. Let the quaternion associated with a spatial rotation R is constructed from its rotation axis S and the rotation angle φ this axis. The associated quaternion is given by,