View source for Applications of dual quaternions to 2D geometry

{{Short description|Four-dimensional algebra over the real numbers}}
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In this article, we discuss certain applications of the [[Dual quaternion|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 a field|algebra]] over the [[Real number|real number]]s.<ref>{{Citation | last1=Matsuda|first1=Genki|title=Anti-commutative Dual Complex Numbers and 2D Rigid Transformation|date=2014| work=Mathematical Progress in Expressive Image Synthesis I: Extended and Selected Results from the Symposium MEIS2013 | pages=131–138 | editor-last=Anjyo|editor-first=Ken|series=Mathematics for Industry|publisher=Springer Japan|language=en | doi=10.1007/978-4-431-55007-5_17|isbn=9784431550075|last2=Kaji|first2=Shizuo|last3=Ochiai|first3=Hiroyuki|arxiv=1601.01754|s2cid=2173557 }}</ref><ref>Gunn C. (2011) On the Homogeneous Model of Euclidean Geometry. In: Dorst L., Lasenby J. (eds) Guide to Geometric Algebra in Practice. Springer, London</ref> Their primary application is in representing rigid body motions in 2D space.

Unlike multiplication of [[Dual number|dual number]]s or of [[Complex number|complex number]]s, that of planar quaternions is non-commutative.

== Definition ==
In this article, the set of planar quaternions is denoted <math>\mathbb {DC}</math>.  A general element <math>q</math> of <math>\mathbb {DC}</math> has the form <math display="inline">A + Bi + C\varepsilon j + D\varepsilon k</math> where <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> are real numbers; <math>\varepsilon</math> is a [[Dual number|dual number]] that squares to zero; and <math>i</math>, <math>j</math>, and <math>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 display="inline"> \varepsilon </math> is [[Nilpotent|nilpotent]] of index <math>2</math>, i.e., <math display="inline"> \varepsilon^2 = 0 </math>, which in some circumstances makes <math display="inline">\varepsilon</math> comparable to an [[Infinitesimal|infinitesimal]] number. It follows that the multiplicative inverses of planar quaternions are given by
<math display="block"> (A + Bi + C\varepsilon j + D\varepsilon k)^{-1} = \frac{A - Bi - C\varepsilon j - D\varepsilon k}{A^2+B^2}</math>

The set <math>\{1, i, \varepsilon j, \varepsilon k\}</math> forms a basis of the vector space of planar quaternions, where the scalars are real numbers.

The magnitude of a planar quaternion <math>q</math> is defined to be <math display="block">|q| = \sqrt{A^2 + B^2}.</math>

For applications in computer graphics, the number <math>A + Bi + C\varepsilon j + D\varepsilon k</math> is commonly represented as the 4-[[Tuple|tuple]] <math>(A,B,C,D)</math>.

== Matrix representation ==
A planar quaternion <math>q = A + Bi + C\varepsilon j + D\varepsilon k</math> has the following representation as a 2x2 complex matrix:
<math display="block">\begin{pmatrix}A + Bi & C + Di \\ 0 & A - Bi \end{pmatrix}.</math>

It can also be represented as a 2×2 dual number matrix:
<math display="block">\begin{pmatrix}A + C\varepsilon & -B + D\varepsilon \\ B + D\varepsilon & A - C\varepsilon\end{pmatrix}.</math>
The above two matrix representations are related to the [[Möbius transformation|Möbius transformations]] and [[Laguerre transformations]] 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|tensor product of algebras]] is commutative up to isomorphism). This algebra can be denoted as <math>\mathbb C[x]/(x^2)</math> using [[Quotient ring|ring quotienting]]. The resulting algebra has a commutative product and is not discussed any further.

== Representing rigid body motions ==
Let <math display="block">q = A + Bi + C\varepsilon j + D\varepsilon k</math> be a unit-length planar quaternion, i.e. we must have that <math display="block">|q| = \sqrt{A^2 + B^2} = 1.</math>

The Euclidean plane can be represented by the set <math display="inline">\Pi = \{i + x \varepsilon j + y \varepsilon k \mid x \in \Reals, y \in \Reals\}</math>.

An element <math>v = i + x \varepsilon j + y \varepsilon k</math> on <math>\Pi</math> represents the point on the [[Euclidean plane]] with Cartesian coordinate <math>(x,y)</math>.

<math>q</math> can be made to [[Group action (mathematics)|act]] on <math>v</math> by <math display="block">qvq^{-1},</math> which maps <math>v</math> onto some other point on <math>\Pi</math>.

We have the following (multiple) polar forms for <math>q</math>:

# When <math>B \neq 0</math>, the element <math>q</math> can be written as <math display="block">\cos(\theta/2) + \sin(\theta/2)(i + x\varepsilon j + y\varepsilon k),</math> which denotes a rotation of angle <math>\theta</math> around the point <math>(x,y)</math>.
# When <math>B = 0</math>, the element <math>q</math> can be written as <math display="block">\begin{aligned}&1 + i(x\varepsilon j + y\varepsilon k)\\ = {} & 1 - y\varepsilon j + x\varepsilon k,\end{aligned}</math> which denotes a translation by vector <math>\begin{pmatrix}x \\ y\end{pmatrix}.</math>

== Geometric construction ==
A principled construction of the planar quaternions can be found by first noticing that they are a subset of the [[Dual quaternion|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 [[Dual quaternion|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 [[Dual quaternion|adequately done elsewhere]].
* The dual quaternions can be understood as an "infinitesimal thickening" of the quaternions.<ref>{{Cite web| url=https://terrytao.wordpress.com/2011/03/05/lines-in-the-euclidean-group-se2/|title=Lines in the Euclidean group SE(2)| date=2011-03-06|website=What's new|access-date=2019-05-28}}</ref><ref>{{Cite journal|last=Study|first=E.|date=December 1891| title=Von den Bewegungen und Umlegungen|journal=Mathematische Annalen|volume=39|issue=4|pages=441–565| doi=10.1007/bf01199824|s2cid=115457030 | issn=0025-5831}}</ref><ref>{{Cite journal|last=Sauer|first=R.|date=1939|title=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.|journal=ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik|volume=19|issue=2| pages=127| doi=10.1002/zamm.19390190222|issn=0044-2267|bibcode=1939ZaMM...19R.127S}}</ref> Recall that the quaternions can be used to represent [[Quaternions and spatial rotation|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 <math>\Pi</math> denote an infinitesimal plane lying on the unit sphere, equal to <math>\{i + x \varepsilon j + y \varepsilon k \mid x \in \mathbb R, y \in \mathbb R\}</math>. Observe that <math>\Pi</math> is a subset of the sphere, in spite of being flat (this is thanks to the behaviour of dual number infinitesimals). {{pb}} Observe then that as a subset of the dual quaternions, the planar quaternions rotate the plane <math>\Pi</math> back onto itself. The effect this has on <math>v \in \Pi</math>  depends on the value of <math>q = A + Bi + C\varepsilon j + D\varepsilon k</math> in <math>qvq^{-1}</math>:
*# When <math>B\neq 0</math>, the axis of rotation points towards some point <math>p</math> on <math>\Pi</math>, so that the points on <math>\Pi</math> experience a rotation around <math>p</math>.
*# When <math>B = 0</math>, the axis of rotation points away from the plane, with the angle of rotation being infinitesimal. In this case, the points on <math>\Pi</math> experience a translation.

== See also ==
* [[Biography:Eduard Study|Eduard Study]]
* [[Quaternion]]
* [[Dual number]]
* [[Dual quaternion]]
* [[Clifford algebra]]
* [[Euclidean plane isometry]]
* [[Affine transformation]]
* [[Projective plane]]
* [[Homogeneous coordinates]]
* SLERP
* [[Conformal geometric algebra]]

== References ==
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{{Number systems}}

[[Category:Hypercomplex numbers]]
[[Category:Quaternions]]
[[Category:Euclidean plane geometry]]
[[Category:Euclidean symmetries]]
[[Category:Clifford algebras]]

{{Sourceattribution|Applications of dual quaternions to 2D geometry}}