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

Revision as of 02:08, 16 June 2021

Revision as of 22:25, 30 June 2021

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== Matrix representation ==

A dual-complex number <math>q=A + Bi + C\varepsilon j + D\varepsilon k</math> has the following representation as a 2x2 complex matrix:

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== 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 dual-complex number, i.e. we must have that <math display="block">|q| = \sqrt{A^2 + B^2} = 1.</math>

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<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>.

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== See also ==

* [[Eduard Study]]

* [[QuaternionsQuaternion]]s

* [[Dual numbersnumber]]s

* [[Dual quaternionsquaternion]]s

* [[Clifford algebra]]

* [[Euclidean plane isometry]]