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 ''dual-complex numbers''. BeIts awareprimary thatapplication theis termin "dual-complexrepresenting numbers"[[rigid often may refer to a different algebra.body Seemotion|rigid discussionsbody onmotions]] thein terminology2D belowspace.
The '''dual-complex numbers''' make up a four-dimensional [[Algebra over a field|algebra]] over the [[real number]]s.<ref>{{Citation | last=Matsuda|first=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}}</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 dual-complex numbers is [[non-commutative]].
