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The usefulness of quaternions for geometrical computations can be generalised to other dimensions by identifying the quaternions as the even part , + of the Clifford algebra , (). This is an associative multivector algebra built up from fundamental basis elements σ 1 , σ 2 , σ 3 using the product rules
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.
Then the quaternion algebra provided the foundation for spherical trigonometry introduced in chapter 9. Hoüel replaced Hamilton's basis vectors i , j , k with i 1 , i 2 , and i 3 . The variety of fonts available led Hoüel to another notational innovation: A designates a point, a and a are algebraic quantities, and in the equation for a quaternion
The Cayley–Dickson construction is due to Leonard Dickson in 1919 showing how the octonions can be constructed as a two-dimensional algebra over quaternions.In fact, starting with a field F, the construction yields a sequence of F-algebras of dimension 2 n.
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:
Each quaternion has a tensor, which is a measure of its magnitude (in the same way as the length of a vector is a measure of a vectors' magnitude). When a quaternion is defined as the quotient of two vectors, its tensor is the ratio of the lengths of these vectors.
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]
A dual quaternion can be represented in the form A + εB, where A and B are ordinary quaternions and ε is the dual unit, which satisfies ε 2 = 0 and commutes with every element of the algebra. Unlike quaternions, the dual quaternions do not form a division algebra.