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The dual-complex numbers (here denoted ${\displaystyle \mathbb {DC} }$) 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 "dual-complex numbers" may be misleading. Unlike the dual numbers or the complex numbers, the dual-complex numbers are non-commutative.

Definition

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

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

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

For applications in computer graphics, the number ${\displaystyle A+Bi+C\epsilon j+D\epsilon k}$ should be represented as the 4-tuple ${\displaystyle (A,B,C,D)}$.

Representing rigid body motions

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

The Euclidean plane can be represented by the set ${\displaystyle \Pi =\{i+x\epsilon j+y\epsilon k\mid x\in \mathbb {R} ,y\in \mathbb {R} \}}$.

An element ${\displaystyle v=i+x\epsilon j+y\epsilon k}$ on ${\displaystyle \Pi }$ represents the point on the Euclidean plane with cartesian coordinate ${\displaystyle (x,y)}$.

${\displaystyle q}$ can be made to act on ${\displaystyle v}$ by $${\displaystyle qvq^{-1},}$$ which maps ${\displaystyle v}$ onto some other point on ${\displaystyle \Pi }$.

We have the following (multiple) polar forms for ${\displaystyle q}$:

1. When ${\displaystyle B\neq 0}$, the element ${\displaystyle q}$ can be written as $${\displaystyle \cos(\theta /2)+\sin(\theta /2)(i+x\epsilon j+y\epsilon k),}$$ which denotes a rotation of angle ${\displaystyle \theta }$ around the point ${\displaystyle (x,y)}$.
2. When ${\displaystyle B=0}$, the element ${\displaystyle q}$ can be written as $${\displaystyle {\begin{aligned}&1+i(x\epsilon j+y\epsilon k)\\&=1-y\epsilon j+x\epsilon k,\end{aligned}}}$$ which denotes a translation by vector ${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}.}$

Geometric construction

A principled construction of the dual-complex numbers can be found by first noticing 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 to represent a subset of those rigid-body motions. This requires some familiarity with the way the dual quaternions act on Euclidean space. We won't describe this approach here as it's adequately done elsewhere.
• As an "infinitesimal thickening" of the quaternions. Recall that the quaternions can be used to represent 3D spatial rotations, while the dual numbers can be used to represent "infinitesimals". Combining those features together allows for rotations to be varied infinitesimally. Let ${\displaystyle \Pi }$ denote an infinitesimal plane lying on the unit sphere, equal to ${\displaystyle \{i+x\epsilon j+y\epsilon k\mid x\in \mathbb {R} ,y\in \mathbb {R} \}}$. Observe that ${\displaystyle \Pi }$ is a subset of the sphere, in spite of being flat (this is thanks to the behaviour of dual number infinitesimals).

Observe then that as a subset of the dual quaternions, the dual complex numbers rotate the plane ${\displaystyle \Pi }$ back onto itself. The effect this has on ${\displaystyle v\in \Pi }$ depends on the value of ${\displaystyle q=A+Bi+C\epsilon j+D\epsilon k}$ in ${\displaystyle qvq^{-1}}$:

1. When ${\displaystyle B\neq 0}$, the axis of rotation points towards some point ${\displaystyle p}$ on ${\displaystyle \Pi }$, so that the points on ${\displaystyle \Pi }$ experience a rotation around ${\displaystyle p}$.
2. When ${\displaystyle B=0}$, the axis of rotation points away from the plane, with the angle of rotation being infinitesimal. In this case, the points on ${\displaystyle \Pi }$ experience a translation.

References

1. Ochiai, Hiroyuki; Kaji, Shizuo; Matsuda, Genki (2016-01-08). "Anti-commutative Dual Complex Numbers and 2D Rigid Transformation". arXiv:1601.01754v1 .
2. Gunn, Charles (2011-01-20). "On the Homogeneous Model Of Euclidean Geometry". arXiv:1101.4542v3 .
3. "geometry - Using dual complex numbers for combined rotation and translation". Mathematics Stack Exchange. Retrieved 2019-05-27. {{cite web}}: Cite has empty unknown parameter: |dead-url= (help)
4. "Lines in the Euclidean group SE(2)". What's new. 2011-03-06. Retrieved 2019-05-28.
5. Study, E. (December 1891). "Von den Bewegungen und Umlegungen". Mathematische Annalen. 39 (4): 441–565. doi:10.1007/bf01199824. ISSN 0025-5831.
6. Sauer, R. (1939). "Dr. Wilhelm Blaschke, Prof. a. d. Universität Hamburg, Ebene Kinematik, eine Vorlesung (Hamburger Math. Einzelschriften, 25. Heft, 1938). 56 S. m. 19 Abb. Leipzig-Berlin 1938, Verlag B. G. Teubner. Preis br. 4 M.". ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik. 19 (2): 127. Bibcode:1939ZaMM...19R.127S. doi:10.1002/zamm.19390190222. ISSN 0044-2267.

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