Dual-complex number
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In this article, we discuss certain applications of the dual quaternion algebra to 2D geometry. At this present time, the article is focused on a 4-dimensional subalgebra of the dual quaternions which we will call the ''planar quaternions''. The planar quaternions 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 planar quaternions is non-commutative.


Definition

In this article, the set of planar quaternions is denoted \mathbb . A general element q of \mathbb has the form A + Bi + C\varepsilon j + D\varepsilon k where A, B, C and D are real numbers; \varepsilon is a dual number that squares to zero; and i, j, and 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 \varepsilon is nilpotent of index 2, i.e., \varepsilon^2 = 0 , which in some circumstances makes \varepsilon comparable to an
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
number. It follows that the multiplicative inverses of planar quaternions are given by (A + Bi + C\varepsilon j + D\varepsilon k)^ = \frac The set \ forms a basis of the vector space of planar quaternions, where the scalars are real numbers. The magnitude of a planar quaternion q is defined to be , q, = \sqrt. For applications in computer graphics, the number A + Bi + C\varepsilon j + D\varepsilon k is commonly represented as the 4- tuple (A,B,C,D).


Matrix representation

A planar quaternion q = A + Bi + C\varepsilon j + D\varepsilon k has the following representation as a 2x2 complex matrix: \beginA + Bi & C + Di \\ 0 & A - Bi \end. It can also be represented as a 2×2 dual number matrix: \beginA + C\varepsilon & -B + D\varepsilon \\ B + D\varepsilon & A - C\varepsilon\end. The above two matrix representations are related to the
Möbius transformations Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * Paul ...
and
Laguerre transformations Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigat ...
respectively.


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: # The dual numbers, but with complex-number entries # 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 \mathbb C (x^2) using ring quotienting. The resulting algebra has a commutative product and is not discussed any further.


Representing rigid body motions

Let q = A + Bi + C\varepsilon j + D\varepsilon k be a unit-length planar quaternion, i.e. we must have that , q, = \sqrt = 1. The Euclidean plane can be represented by the set \Pi = \. An element v = i + x \varepsilon j + y \varepsilon k on \Pi represents the point on the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
with Cartesian coordinate (x,y). q can be made to act on v by qvq^, which maps v onto some other point on \Pi. We have the following (multiple) polar forms for q: # When B \neq 0, the element q can be written as \cos(\theta/2) + \sin(\theta/2)(i + x\varepsilon j + y\varepsilon k), which denotes a rotation of angle \theta around the point (x,y). # When B = 0, the element q can be written as \begin&1 + i(x\varepsilon j + y\varepsilon k)\\ = & 1 - y\varepsilon j + x\varepsilon k,\end which denotes a translation by vector \beginx \\ y\end.


Geometric construction

A principled construction of the planar quaternions 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 planar quaternions on the plane: * As a way to represent rigid body motions in 3D space. The planar quaternions 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. 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 \Pi denote an infinitesimal plane lying on the unit sphere, equal to \. Observe that \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 planar quaternions rotate the plane \Pi back onto itself. The effect this has on v \in \Pi depends on the value of q = A + Bi + C\varepsilon j + D\varepsilon k in qvq^: *# When B\neq 0, the axis of rotation points towards some point p on \Pi, so that the points on \Pi experience a rotation around p. *# 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.


See also

* Eduard Study *
Quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
* Dual number * Dual quaternion *
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
* Euclidean plane isometry *
Affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
* Projective plane *
Homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
*
SLERP In computer graphics, Slerp is shorthand for spherical linear interpolation, introduced by Ken Shoemake in the context of quaternion interpolation for the purpose of animation, animating 3D rotation. It refers to constant-speed motion along a unit ...
* Conformal geometric algebra


References

{{Number systems Hypercomplex numbers Quaternions Euclidean plane geometry Euclidean symmetries Clifford algebras