Dual-complex number

The dual-complex numbers make up a four-dimensional algebra over the real numbers.[1][2] Their primary application is in representing rigid body motions in 2D space.

Dual-complex multiplication

${\displaystyle \times }$	${\displaystyle 1}$	${\displaystyle i}$	${\displaystyle \varepsilon j}$	${\displaystyle \varepsilon k}$
${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle i}$	${\displaystyle \varepsilon j}$	${\displaystyle \varepsilon k}$
${\displaystyle i}$	${\displaystyle i}$	${\displaystyle -1}$	${\displaystyle \varepsilon k}$	${\displaystyle -\varepsilon j}$
${\displaystyle \varepsilon j}$	${\displaystyle \varepsilon j}$	${\displaystyle -\varepsilon k}$	${\displaystyle 0}$	${\displaystyle 0}$
${\displaystyle \varepsilon k}$	${\displaystyle \varepsilon k}$	${\displaystyle \varepsilon j}$	${\displaystyle 0}$	${\displaystyle 0}$

Unlike multiplication of dual numbers or of complex numbers, that of dual-complex numbers is non-commutative.

Definition

In this article, the set of dual-complex numbers is denoted ${\displaystyle \mathbb {DC} }$. A general element ${\displaystyle q}$ of ${\displaystyle \mathbb {DC} }$ has the form ${\textstyle A+Bi+C\varepsilon j+D\varepsilon k}$ where ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$ and ${\displaystyle D}$ are real numbers; ${\displaystyle \varepsilon }$ is a dual number that squares to zero; and ${\displaystyle i}$, ${\displaystyle j}$, and ${\displaystyle k}$ are the standard basis elements of the quaternions.

Multiplication is done in the same way as with the quaternions, but with the additional rule that ${\textstyle \varepsilon }$ is nilpotent of index ${\displaystyle 2}$, i.e. ${\textstyle \varepsilon ^{2}=0}$. It follows that the multiplicative inverses of dual-complex numbers are given by

${\displaystyle (A+Bi+C\varepsilon j+D\varepsilon k)^{-1}={\frac {A-Bi-C\varepsilon j-D\varepsilon k}{A^{2}+B^{2}}}}$

The set ${\displaystyle \{1,i,\varepsilon j,\varepsilon k\}}$ forms a basis of the vector space of dual-complex numbers, where the scalars are real 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 ${\displaystyle A+Bi+C\varepsilon j+D\varepsilon k}$ should be represented as the 4-tuple ${\displaystyle (A,B,C,D)}$.

Matrix representation

A dual-complex number ${\displaystyle q=A+Bi+C\varepsilon j+D\varepsilon k}$ has the following representation as a 2x2 complex matrix:

${\displaystyle {\begin{pmatrix}A+Bi&C+Di\\0&A-Bi\end{pmatrix}}.}$

It can also be represented as a 2x2 dual number matrix:

${\displaystyle {\begin{pmatrix}A+C\epsilon &-B+D\epsilon \\B+D\epsilon &A-C\epsilon \end{pmatrix}}.}$

Terminology

The algebra discussed in this article is sometimes called the dual complex numbers. This may be a misleading name because it suggests that the algebra should take the form of either:

1. The dual numbers, but with complex number entries
2. The complex numbers, but with dual number entries

An algebra meeting either description exists. And both descriptions are equivalent. (This is due to the fact that the tensor product of algebras is commutative up to isomorphism). This algebra can be denoted as ${\displaystyle \mathbb {C} [x]/(x^{2})}$ using ring quotienting. The resulting algebra has a commutative product and is not discussed any further.

Representing rigid body motions

Let

$${\displaystyle q=A+Bi+C\varepsilon j+D\varepsilon k}$$

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 ${\textstyle \Pi =\{i+x\varepsilon j+y\varepsilon k\mid x\in \mathbb {R} ,y\in \mathbb {R} \}}$.

An element ${\displaystyle v=i+x\varepsilon j+y\varepsilon k}$ 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
   $${\displaystyle \cos(\theta /2)+\sin(\theta /2)(i+x\varepsilon j+y\varepsilon k),}$$

   
   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
   $${\displaystyle {\begin{aligned}&1+i(x\varepsilon j+y\varepsilon k)\\={}&1-y\varepsilon j+x\varepsilon k,\end{aligned}}}$$

   
   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 are 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 will not describe this approach here as it is adequately done elsewhere.
• The dual quaternions can be understood as an "infinitesimal thickening" of the quaternions.[3][4][5] 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 ${\displaystyle \{i+x\varepsilon j+y\varepsilon k\mid x\in \mathbb {R} ,y\in \mathbb {R} \}}$. 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 ${\displaystyle q=A+Bi+C\varepsilon j+D\varepsilon k}$ 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.

See also

• Eduard Study
• Quaternions
• Dual numbers
• Dual quaternions
• Clifford algebra
• Euclidean plane isometry
• Affine transformation
• Projective plane
• Homogeneous coordinates
• SLERP
• Conformal geometric algebra

References

1. Matsuda, Genki; Kaji, Shizuo; Ochiai, Hiroyuki (2014), Anjyo, Ken (ed.), "Anti-commutative Dual Complex Numbers and 2D Rigid Transformation", Mathematical Progress in Expressive Image Synthesis I: Extended and Selected Results from the Symposium MEIS2013, Mathematics for Industry, Springer Japan, pp. 131–138, arXiv:1601.01754, doi:10.1007/978-4-431-55007-5_17, ISBN 9784431550075
2. Gunn C. (2011) On the Homogeneous Model of Euclidean Geometry. In: Dorst L., Lasenby J. (eds) Guide to Geometric Algebra in Practice. Springer, London
3. "Lines in the Euclidean group SE(2)". What's new. 2011-03-06. Retrieved 2019-05-28.
4. Study, E. (December 1891). "Von den Bewegungen und Umlegungen". Mathematische Annalen. 39 (4): 441–565. doi:10.1007/bf01199824. ISSN 0025-5831.
5. 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.