Applications of dual quaternions to 2D geometry: Difference between revisions

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{{Short description|Four-dimensional algebra over the real numbers}}

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The '''dual-complexplanar numbersquaternions''' make up a four-dimensional [[Algebra over a field|algebra]] over the [[real number]]s.<ref>{{Citation |last last1=Matsuda|firstfirst1=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., [[Joan Lasenby|Lasenby J.]] (eds) Guide to Geometric Algebra in Practice. Springer, London</ref> Their primary application is in representing [[rigid body motion|rigid body motions]] in 2D space. In this article, certain applications of the [[dual quaternion]] algebra to 2D geometry are discussed. At this present time, the article is focused on a 4-dimensional subalgebra of the dual quaternions which will later be called the ''planar quaternions''.

Unlike multiplication of [[dual number]]s or of [[complex number]]s, that of dual-complex numbers is [[non-commutative]].

Unlike multiplication of [[dual number]]s or of [[complex number]]s, that of dual-complexplanar numbersquaternions is [[non-commutative]].

== Definition ==

In this article, the set of dual-complexplanar numbersquaternions 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]] 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]] 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]] number. It follows that the multiplicative inverses of dual-complexplanar numbersquaternions 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, (A + Bi +i, C\varepsilon j, + D\varepsilon k)^{-1\}</math> =forms \frac{Aa -basis Biof -C\varepsilonthe jvector -space D\varepsilonof planar quaternions, k}{A^2+B^2}</math>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>

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

TheFor magnitudeapplications ofin acomputer dual-complexgraphics, the number <math>qA + Bi + C\varepsilon j + D\varepsilon k</math> is definedcommonly torepresented beas the 4-[[tuple]] <math display="block">|q| = \sqrt{(A^2 + ,B^2}.,C,D)</math>.

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

== Matrix representation ==

A dual-complexplanar numberquaternion <math>q = A + Bi + C\varepsilon j + D\varepsilon k</math> has the following representation as a [[Mobius transformation|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:

A dual-complex number <math>q=A + Bi + C\varepsilon j + D\varepsilon k</math> has the following representation as a [[Mobius transformation|2x2 complex 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.

: <math>\begin{pmatrix}A + Bi & C + Di \\ 0 & A - Bi \end{pmatrix}</math>

== 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 <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>

LetThe Euclidean plane can be represented by the set <math display="blockinline">q\Pi = A\{i + Bi +x C\varepsilon j + Dy \varepsilon k</math> be\mid ax unit-length\in dual-complex number\Reals, i.e.y we must have that <math display="block">|q| =\in \sqrt{A^2 + B^2Reals\} = 1.</math>.

TheAn Euclidean plane can be represented by the setelement <math display="inline">\Piv = \{i + x \varepsilon j + y \varepsilon k</math> on <math>\midPi</math> xrepresents \inthe \mathbbpoint R,on ythe \in[[Euclidean \mathbbplane]] R\}with [[Cartesian coordinate]] <math>(x,y)</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) [[Polarpolar form|polar forms]]s 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\left(\frac{\Delta x}{2} \varepsilon j + \frac{\Delta y}{2}\varepsilon k\right)\\ = {} & 1 - \frac{\Delta y}{2}\varepsilon j + \frac{\Delta x}{2}\varepsilon k,\end{aligned}</math> which denotes a translation by vector <math>\begin{pmatrix}\Delta x \\ \Delta y\end{pmatrix}.</math>

== Geometric construction ==

A principled construction of the dual-complexplanar numbersquaternions 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 dual-complexplanar numbersquaternions on the plane:

* As a way to represent [[Dual quaternion|rigid body motions in 3D space]]. The dual-complexplanar numbersquaternions 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.

: Observe then that as a subset of the dual quaternions, the dual complex numbers 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 ==

* [[Eduard Study]]

* [[QuaternionsQuaternion]]

* [[Dual numbersnumber]]

* [[Dual quaternionsquaternion]]

* [[Clifford algebra]]

* [[Euclidean plane isometry]]