Quaternions

Quaternions are numbers of the form $a+bi+cj+dk$, where $a$, $b$, $c$ and $d$ are real numbers and each of $i,j,k$ denotes a number-like entity satisfying $i^{2}=-1$, $j^{2}=-1$ and $k^{2}=-1$ respectively. Of course, since the square of any real number is nonnegative, none of the entities $i$, $j$ or $k$ can be a real number. At first glance, it is not even clear whether such objects can exist in any meaningful sense: for example, can we sensibly associate with $i$, $j$ and $k$ natural operations such as addition and multiplication? As it happens, we can define such mathematical operations in a consistent and sensible way and, perhaps more importantly, the resulting system provides mathematicians, physicists, and engineers with a powerful approach to expressing parts of these sciences in a convenient and natural-feeling way.

Historical context

The need for quaternions became appearent after the sucessful introduction of complex numbers into mathematics. These numbers made it possible to add, subtract, multiply and divide tuplets - points in the plane - just like one can do with real numbers. The search for a larger system, where one can similarly deal with triplets - points in 3-dimensional space - became the natural next step. It turned out, however, that there is no way triplets of real numbers can form such a system. The breakthrough came with Sir William Rowan Hamilton, when he realized that quadruples would work. He famously inscribed their defining equation on Broom Bridge in Dublin when walking with his wife on 16 October 1843.

Working with quaternions

Quaternion arithmetic is surprisingly straightforward. The main difficulty is that commutativity in multiplication is lost - for two quaterions q₁ and q₂ we cannot, in general, expect that q₁q₂=q₂q₁.

Basic operations

Quaternion addition is straightforward,

$(a_{0}+b_{0}i+c_{0}j+d_{0}k)+(a_{1}+b_{1}i+c_{1}j+d_{1}k)=a_{0}+a_{1}+(b_{0}+b_{1})i+(c_{0}+c_{1})j+(c_{0}+c_{1})k$.

The result is again a quaternion.

Multiplication is a little more complicated.

Definition & basic operations

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

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

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

They are a non-commutative extension of the real 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.

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
• SU2
• SO3

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