Applications of dual quaternions to 2D geometry

The dual-complex numbers (here denoted $\mathbb {DC}$) is a 4-dimensional algebra over the real numbers. Its primary application is in representing rigid body motions in 2D space.

Beware that the term may be misleading. Unlike the dual numbers or the complex numbers, the dual-complex numbers are non-commutative.

Definition

A general element $q$ of $\mathbb {DC}$ is defined to be$$A+Bi+C\epsilon j+D\epsilon k$$where $A$, $B$, $C$ and $D$ are arbitrary real numbers; $\epsilon$ is a dual number that squares to zero; and $i$, $j$ and $k$ are the standard basis elements of the quaternions.

The set $\{1,i,\epsilon j,\epsilon k\}$ forms a basis of the dual-complex numbers.

The magnitude of a dual quaternion $q$ is defined to be $$|q|={\sqrt {A^{2}+B^{2}}}.$$

Representing rigid body motions

Let $$q=A+Bi+C\epsilon j+D\epsilon k$$ be a unit dual-complex number, i.e. $$|q|={\sqrt {A^{2}+B^{2}}}=1.$$

The element $q$ acts on points in the plane by $$qvq^{-1}$$ where $$v=i+x\epsilon j+y\epsilon k$$ represents the point with Cartesian coordinate $(x,y)$. Note that $v$ can also be understood as the homogeneous coordinate $(1,x,y)$.

We have the following polar forms for $q$:

When $B\neq 0$, the element $q$ can be written as $$\cos(\theta /2)+\sin(\theta /2)(i+x\epsilon j+y\epsilon k),$$ which denotes a rotation of angle $\theta$ around point $(x,y)$.

When $B=0$, the element $q$ can be written as $$1+i(x\epsilon j+y\epsilon k),$$ which denotes a translation by vector ${\begin{pmatrix}x\\y\end{pmatrix}}.$

Geometric construction

A principled construction of the dual-complex numbers can be found by seeing that they're a subset of the dual-quaternions.

There are two geometric interpretations of the dual-quaternions, both of which can be used to derive the action of the dual-complex numbers on the plane:

• As a way to represent rigid body motions in 3D space. The dual-complex numbers can then be seen as representing a subset of those rigid-body motions. This requires some familiarity with the way that the dual quaternions act on Euclidean space.
• As an "infinitesimal thickening" of the quaternions. Note that the quaternions are often used to express rotations in 3D space; and the dual numbers are often used to represent "infinitesimals". The combination of the two allows rotations to be varied infinitesimally. Let $\Pi$ denote an infinitesimal plane lying on the unit sphere, precisely equal to $\{i+x\epsilon j+y\epsilon k\mid x\in \mathbb {R} ,y\in \mathbb {R} \}$. Observe that $\Pi$ is a subset of the sphere, in spite of being flat; this is thanks to the behaviour of dual number infinitesimals.

Notice that as a subset of the dual quaternions, the dual complex numbers rotate the plane $\Pi$ back to itself. When $B\neq 0$ (see the general form of a dual-complex number), the axis of rotation points towards some point on $\Pi$, so that points on $\Pi$ experience a regular rotation. When $B=0$, the axis of rotation points away from the plane, with the angle of rotation being infinitesimal; in this case, the points on $\Pi$ experience a translation.