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

Quaternions can be used to model the three-dimensional rotation group. First, every 3-dimensional vector $(x,y,z)$ is associated with the quaternion $ix+jy+kz$. A rotation over an angle $\alpha$ around the axis defined by the unit vector $u$ is then represented by the unit quaternion

$\cos \left({\frac {\alpha }{2}}\right)+u\sin \left({\frac {\alpha }{2}}\right).$

In other words, the image of a vector under this rotation is the same as the product of the quaternion representing the rotation with the quaternion representing the vector.

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

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