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

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

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

Applications

Quaternions can be used to model the three-dimensional rotation group. Given a 3-dimensional unit vector u and an angle $alpha$, the unit quaternion $\cos({\frac {\alpha }{2}})+u*\sin({\frac {\alpha }{2}})$ can be used to represent a rotation of $\alpha$ around the axis defined by u.

The set of such unit quaternions form a group under quaternion multiplication.

References

• Henry Baker. Henry Bakers quaternion page. Electronic document.

• Simon Altmann (2005). Rotations, Quaternions, and Double Groups. Dover Publications. ISBN-10: 0486445186. ISBN-13: 978-0486445182. (First edition appeared in 1977).