Dual-complex number

The dual-complex numbers make up a four-dimensional algebra over the real numbers. Their primary application is in representing rigid body motions in 2D space.
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 formula_1. A general element formula_2 of formula_1 has the form formula_4 where formula_5, formula_6, formula_7 and formula_8 are real numbers; formula_9 is a dual number that squares to zero; and formula_10, formula_11, and formula_12 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 formula_13 is nilpotent of index formula_14, i.e. formula_15, which in some circumstances makes formula_16 comparable to an infinitesimal number. It follows that the multiplicative inverses of dual-complex numbers are given by
The set formula_18 forms a basis of the vector space of dual-complex numbers, where the scalars are real numbers.
The magnitude of a dual-complex number formula_2 is defined to be formula_20
For applications in computer graphics, the number formula_21 is commonly represented as the 4-tuple formula_22.
Matrix representation.
A dual-complex number formula_23 has the following representation as a 2x2 complex matrix:
It can also be represented as a 2x2 dual number matrix:
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:
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 formula_26 using ring quotienting. The resulting algebra has a commutative product and is not discussed any further.
Representing rigid body motions.
Let formula_27 be a unit-length dual-complex number, i.e. we must have that formula_28
The Euclidean plane can be represented by the set formula_29.
An element formula_30 on formula_31 represents the point on the Euclidean plane with cartesian coordinate formula_32.
formula_2 can be made to act on formula_34 by formula_35 which maps formula_34 onto some other point on formula_31.
We have the following (multiple) polar forms for formula_2:
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: