Applications of dual quaternions to 2D geometry

(Redirected from Dual-complex numbers)
Planar quaternion multiplication

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.[1][2] 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

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In this article, the set of planar quaternions is denoted  . A general element   of   has the form   where  ,  ,   and   are real numbers;   is a dual number that squares to zero; and  ,  , and   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   is nilpotent of index  , i.e.,  , which in some circumstances makes   comparable to an infinitesimal number. It follows that the multiplicative inverses of planar quaternions are given by  

The set   forms a basis of the vector space of planar quaternions, where the scalars are real numbers.

The magnitude of a planar quaternion   is defined to be  

For applications in computer graphics, the number   is commonly represented as the 4-tuple  .

Matrix representation

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A planar quaternion   has the following representation as a 2x2 complex matrix:  

It can also be represented as a 2×2 dual number matrix:   The above two matrix representations are related to the Möbius transformations and Laguerre transformations respectively.

Terminology

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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   using ring quotienting. The resulting algebra has a commutative product and is not discussed any further.

Representing rigid body motions

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Let   be a unit-length planar quaternion, i.e. we must have that  

The Euclidean plane can be represented by the set  .

An element   on   represents the point on the Euclidean plane with Cartesian coordinate  .

  can be made to act on   by   which maps   onto some other point on  .

We have the following (multiple) polar forms for  :

  1. When  , the element   can be written as   which denotes a rotation of angle   around the point  .
  2. When  , the element   can be written as   which denotes a translation by vector  

Geometric construction

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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.[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   denote an infinitesimal plane lying on the unit sphere, equal to  . Observe that   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   back onto itself. The effect this has on   depends on the value of   in  :
    1. When  , the axis of rotation points towards some point   on  , so that the points on   experience a rotation around  .
    2. When  , the axis of rotation points away from the plane, with the angle of rotation being infinitesimal. In this case, the points on   experience a translation.

See also

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References

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  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, S2CID 2173557
  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. S2CID 115457030.
  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.