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

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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 a field|algebra]] over the [[real number]]s.<ref>{{Citation | lastlast1=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., 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.

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

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* 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.