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The '''planar quaternions''' make up a four-dimensional [[Algebra over a field|algebra]] over the [[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., [[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''.▼
▲The '''planar quaternions''' make up a four-dimensional [[Algebra over a field|algebra]] over the [[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 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]].
== 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]] that squares to zero; and <math>i</math>, <math>j</math>, and <math>k</math> are the standard basis elements of the [[quaternions]].
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# 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 ==
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