Generalizations

If F is any acreage with appropriate altered from 2, and a and b are elements of F, one may ascertain a four-dimensional unitary akin algebra over F with base 1, i, j, and ij, area i2 = a, j2 = b and ij = −ji (so (ij)2 = −ab). These algebras are alleged quaternion algebras and are isomorphic to the algebra of 2×2 matrices over F or anatomy analysis algebras over F, depending on the best of a and b.

editQuaternions as the even allotment of Cℓ3,0(R)

The account of quaternions for geometrical computations can be generalised to added dimensions, by anecdotic the quaternions as the even allotment Cℓ+3,0(R) of the Clifford algebra Cℓ3,0(R). This is an akin multivector algebra congenital up from axiological base elements σ1, σ2, σ3 application the artefact rules

If these axiological base elements are taken to represent vectors in 3D space, again it turns out that the absorption of a agent r in a even erect to a assemblage agent w can be written:

Two reflections accomplish a circling by an bend alert the bend amid the two absorption planes, so

corresponds to a circling of 180° in the even absolute σ1 and σ2. This is actual agnate to the agnate quaternion formula,

In fact, the two are identical, if we accomplish the identification

and it is aboveboard to affirm that this preserves the Hamilton relations

In this picture, quaternions accord not to vectors but to bivectors, quantities with consequence and orientations associated with accurate 2D planes rather than 1D directions. The affiliation to circuitous numbers becomes clearer, too: in 2D, with two agent admonition σ1 and σ2, there is alone one bivector base aspect σ1σ2, so alone one imaginary. But in 3D, with three agent directions, there are three bivector base elements σ1σ2, σ2σ3, σ3σ1, so three imaginaries.

This acumen extends further. In the Clifford algebra Cℓ4,0(R), there are six bivector base elements, back with four altered basal agent directions, six altered pairs and accordingly six altered linearly absolute planes can be defined. Rotations in such spaces application these generalisations of quaternions, alleged rotors, can be actual advantageous for applications involving constant coordinates. But it is alone in 3D that the amount of base bivectors equals the amount of base vectors, and anniversary bivector can be articular as a pseudovector.

Dorst et al. analyze the afterward advantages for agreement quaternions in this added setting:22

Rotors are accustomed and non-mysterious in geometric algebra and calmly accepted as the encoding of a bifold reflection.

In geometric algebra, a rotor and the altar it acts on reside in the aforementioned space. This eliminates the charge to change representations and to encode new abstracts structures and methods (which is appropriate if assiduity beeline algebra with quaternions).

A rotor is universally applicative to any aspect of the algebra, not just vectors and added quaternions, but aswell lines, planes, circles, spheres, rays, and so on.

In the conformal archetypal of Euclidean geometry, rotors acquiesce the encoding of rotation, adaptation and ascent in a individual aspect of the algebra, universally acting on any element. In particular, this agency that rotors can represent rotations about an approximate axis, admitting quaternions are bound to an arbor through the origin.

Rotor-encoded transformations accomplish departure decidedly straightforward.

For added detail about the geometrical uses of Clifford algebras, see Geometric algebra.