1000/1000
Hot
Most Recent
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.
In this article, the set of dual-complex numbers is denoted [math]\displaystyle{ \mathbb {DC} }[/math]. A general element [math]\displaystyle{ q }[/math] of [math]\displaystyle{ \mathbb {DC} }[/math] has the form [math]\displaystyle{ A + Bi + C\varepsilon j + D\varepsilon k }[/math] where [math]\displaystyle{ A }[/math], [math]\displaystyle{ B }[/math], [math]\displaystyle{ C }[/math] and [math]\displaystyle{ D }[/math] are real numbers; [math]\displaystyle{ \varepsilon }[/math] is a dual number that squares to zero; and [math]\displaystyle{ i }[/math], [math]\displaystyle{ j }[/math], and [math]\displaystyle{ k }[/math] 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 [math]\displaystyle{ \varepsilon }[/math] is nilpotent of index [math]\displaystyle{ 2 }[/math], i.e. [math]\displaystyle{ \varepsilon^2=0 }[/math], which in some circumstances makes [math]\displaystyle{ \varepsilon }[/math] comparable to an infinitesimal number. It follows that the multiplicative inverses of dual-complex numbers are given by
The set [math]\displaystyle{ \{1, i, \varepsilon j, \varepsilon k\} }[/math] forms a basis of the vector space of dual-complex numbers, where the scalars are real numbers.
The magnitude of a dual-complex number [math]\displaystyle{ q }[/math] is defined to be [math]\displaystyle{ |q| = \sqrt{A^2 + B^2}. }[/math]
For applications in computer graphics, the number [math]\displaystyle{ A + Bi + C\varepsilon j + D\varepsilon k }[/math] is commonly represented as the 4-tuple [math]\displaystyle{ (A,B,C,D) }[/math].
A dual-complex number [math]\displaystyle{ q=A + Bi + C\varepsilon j + D\varepsilon k }[/math] has the following representation as a 2x2 complex matrix:
It can also be represented as a 2x2 dual number matrix:
The above two matrix representations are related to the Möbius transformations and Laguerre transformations respectively.
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 [math]\displaystyle{ \mathbb C[x]/(x^2) }[/math] using ring quotienting. The resulting algebra has a commutative product and is not discussed any further.
Let [math]\displaystyle{ q = A + Bi + C\varepsilon j + D\varepsilon k }[/math] be a unit-length dual-complex number, i.e. we must have that [math]\displaystyle{ |q| = \sqrt{A^2 + B^2} = 1. }[/math]
The Euclidean plane can be represented by the set [math]\displaystyle{ \Pi = \{i + x \varepsilon j + y \varepsilon k \mid x \in \mathbb R, y \in \mathbb R\} }[/math].
An element [math]\displaystyle{ v = i + x \varepsilon j + y \varepsilon k }[/math] on [math]\displaystyle{ \Pi }[/math] represents the point on the Euclidean plane with cartesian coordinate [math]\displaystyle{ (x,y) }[/math].
[math]\displaystyle{ q }[/math] can be made to act on [math]\displaystyle{ v }[/math] by [math]\displaystyle{ qvq^{-1}, }[/math] which maps [math]\displaystyle{ v }[/math] onto some other point on [math]\displaystyle{ \Pi }[/math].
We have the following (multiple) polar forms for [math]\displaystyle{ q }[/math]:
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: