Editing Screw theory (section)

== Homography ==
The combination of a translation with a rotation effected by a screw displacement can be illustrated with the [[exponential map (Lie theory)|exponential mapping]].

Since ''&epsilon;''<sup>2</sup> = 0 for [[dual numbers]], exp(''a&epsilon;'') = 1 + ''a&epsilon;'', all other terms of the exponential series vanishing.

Let ''F'' = {1 + ''&epsilon;r'' : ''r'' ∈ '''H'''}, ''&epsilon;''<sup>2</sup> = 0.
Note that ''F'' is [[invariant (mathematics)#Invariant set|stable]] under the [[quaternions and spatial rotation|rotation]] {{nowrap|''q'' → ''p''<sup>−1</sup>''qp''}} and under the translation {{nowrap|1=(1 + ''&epsilon;r'')(1 + ''&epsilon;s'') = 1 + ''&epsilon;''(''r'' + ''s'')}} for any vector quaternions ''r'' and&nbsp;''s''.
''F'' is a [[flat (geometry)|3-flat]] in the eight-dimensional space of [[dual quaternion]]s. This 3-flat ''F'' represents [[space]], and the [[homography]] constructed, [[restriction of a function|restricted]] to ''F'', is a screw displacement of space.

Let ''a'' be half the angle of the desired turn about axis ''r'', and ''br'' half the displacement on the [[screw axis]]. Then form {{nowrap|1=''z'' = exp((''a'' + ''b&epsilon;'')''r'')}} and {{nowrap|1=''z''* = exp((''a'' − ''b&epsilon;'')''r'')}}. Now the homography is
: <math>[q : 1]\begin{pmatrix}z & 0 \\ 0 & z^* \end{pmatrix} = [q z : z^*] \thicksim [(z^*)^{-1} q z : 1].</math>
The inverse for ''z''* is
: <math> \frac 1 {\exp(ar - b \varepsilon r)} = (e^{ar} e^{-br \varepsilon} )^{-1} = e^{br \varepsilon} e^{-ar},</math>
so, the homography sends ''q'' to
: <math>(e^{b \varepsilon} e^{-ar}) q (e^{ar} e^{b \varepsilon r}) = e^{b \varepsilon r} (e^{-ar} q  e^{ar} )e^{b \varepsilon r} = e^{2b \varepsilon r} (e^{-ar} q e^{ar}).</math>

Now for any quaternion vector ''p'', {{nowrap|1=''p''* = −''p''}}, let {{nowrap|1=''q'' = 1 + ''p&epsilon;'' ∈ ''F''}}, where the required rotation and translation are effected.

Evidently the [[group of units]] of the [[ring (mathematics)|ring]] of dual quaternions is a [[Lie group]]. A subgroup has [[Lie algebra]] generated by the parameters ''a r'' and ''b s'', where {{nowrap|''a'', ''b'' ∈ '''R'''}}, and {{nowrap|''r'', ''s'' ∈ '''H'''}}. These six parameters generate a subgroup of the units, the unit sphere. Of course it includes ''F'' and the [[3-sphere]] of [[versor]]s.