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== Geometric construction ==
A principled construction of the dual-complex numbers can be found by first noticing that they
There are two geometric interpretations of the ''dual-quaternions'', both of which can be used to derive the action of the dual-complex numbers on the plane:
* As a way to represent [[Dual quaternion|rigid body motions in 3D space]]. The dual-complex numbers 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
* The dual quaternions can be understood as an "infinitesimal thickening" of the quaternions.<ref>{{Cite web|url=https://math.stackexchange.com/a/3168611/228274|title=geometry - Using dual complex numbers for combined rotation and translation|website=Mathematics Stack Exchange|access-date=2019-05-27}}</ref><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|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).
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