Quaternions

Quaternions are a non-commutative extension of the complex numbers. They were first described by Sir William Rowan Hamilton in 1843. He famously inscribed their defining equation on Broom Bridge in Dublin when walking with his wife on 16 October 1843. They have many possible applications, including in computer graphics, but have during their history proved comparatively unpopular, with vectors being preferred instead.

Definition & basic operations

The quaternions, $\mathbb {H}$, form a four-dimensional normed division algebra over the real numbers.

$\mathbb {H} =\left\lbrace a+{\mathit {i}}b+{\mathit {j}}c+{\mathit {k}}d\mid a,b,c,d\in \mathbb {R} \right\rbrace$

${\mathit {i}}^{2}={\mathit {j}}^{2}={\mathit {k}}^{2}={\mathit {ijk}}=-1\,$

Applications

In 3-dimensional space, any sequence of rotations around any number of different axes intersecting the origin can be represented by a single rotation - the set of all such rotations form a group.

The set of unit quaternions under quaternion multiplication also form a group, which can be used to model the three-dimensional rotation group.

A unit quaternion then represents a rotation, multiplying two quaternions represents performing two rotations in sequence, the resulting quaternion represents the equivalent single rotation.

Given an ordinary 3-dimensional vector u₁ of unit length and an angle $\alpha _{1}$, the quaternion

$q_{1}=\cos \left({\frac {\alpha _{1}}{2}}\right)+u_{1}\sin \left({\frac {\alpha _{1}}{2}}\right)$

then represents a rotation over an angle $\alpha _{1}$ around the axis defined by the unit vector $u_{1}$.

Given a similarly defined quaternion

$q_{2}=\cos \left({\frac {\alpha _{2}}{2}}\right)+u_{2}\sin \left({\frac {\alpha _{2}}{2}}\right)$

one can compute their product quaternion

$q_{1}q_{2}=\cos \left({\frac {\alpha _{1}}{2}}\right)\cos \left({\frac {\alpha _{2}}{2}}\right)-u_{1}\bullet u_{2}+u_{1}\times u_{2}$

This quaternion can be rewritten in the form

$\cos \left({\frac {\alpha _{3}}{2}}\right)+u_{3}\sin \left({\frac {\alpha _{3}}{2}}\right)$.

It represents a rotation over an angle $\alpha _{3}$ around the axis defined by the unit vector $u_{3}$, with

$\cos \left({\frac {\alpha _{3}}{2}}\right)=\cos \left({\frac {\alpha _{1}}{2}}\right)\cos \left({\frac {\alpha _{2}}{2}}\right)-u_{1}\bullet u_{2}$, and

$u_{3}=u_{1}\times u_{2}$

Note that each of the quaternion units (i,j,k) in this model represents a 180 degree rotation, and the quaternion -1 represents a full rotation. The quaternion representation thus keeps track of rotations, in addition to a fermionic phase factor of +-1.

See also

• Division algebra
• Field
• Algebra

Related topics

• Rotation
• Euler angles
• Octonions
• Hypercomplex number

References

• Henry Baker. Quaternion references. Electronic document.
• Simon Altmann (2005). Rotations, Quaternions, and Double Groups. Dover Publications. ISBN 978-0486445182. (First edition appeared in 1977).

External links

• Quaternion at MathWorld