The Laguerre transformations or axial homographies are an analogue of Möbius transformations over the dual numbers. When studying these transformations, the dual numbers are often interpreted as representing oriented lines on the plane. The Laguerre transformations map lines to lines, and include in particular all isometries of the plane. Strictly speaking, these transformations act on the dual number projective line, which adjoins to the dual numbers a set of points at infinity. Topologically, this projective line is equivalent to a cylinder. Points on this cylinder are in a natural one-to-one correspondence with oriented lines on the plane.
Definition A Laguerre transformation is a linear fractional transformation z ↦ a z + b c z + d {\displaystyle z\mapsto {\frac {az+b}{cz+d}}} where a , b , c , d {\displaystyle a,b,c,d} are all dual numbers, z {\displaystyle z} lies on the dual number projective line, and a d − b c {\displaystyle ad-bc} is not a zero divisor. A dual number is a hypercomplex number of the form x + y ε {\displaystyle x+y\varepsilon } where ε 2 = 0 {\displaystyle \varepsilon ^{2}=0} but ε ≠ 0 {\displaystyle \varepsilon \neq 0} . This can be compared to the complex numbers which are of the form x + y i {\displaystyle x+yi} where i 2 = − 1 {\displaystyle i^{2}=-1} . The points of the dual number projective line can be defined equivalently in two ways:
The usual set of dual numbers, but with some additional "points at infinity". Formally, the set is { x + y ε ∣ x ∈ R , y ∈ R } ∪ { 1 x ε ∣ x ∈ R } {\displaystyle \{x+y\varepsilon \mid x\in \mathbb {R} ,y\in \mathbb {R} \}\cup \left\{{\frac {1}{x\varepsilon }}\mid x\in \mathbb {R} \right\}} . The points at infinity can be expressed as 1 x ε {\displaystyle {\frac {1}{x\varepsilon }}} where x {\displaystyle x} is an arbitrary real number. Different values of x {\displaystyle x} correspond to different points at infinity. These points are infinite because ε {\displaystyle \varepsilon } is often understood as being an infinitesimal number, and so 1 / ε {\displaystyle 1/\varepsilon } is therefore infinite. The homogeneous coordinates [x : y] with x and y dual numbers such that the ideal that they generate is the whole ring of dual numbers. The ring is viewed through the injection x ↦ [x : 1]. The projective line includes points [1 : yε].
Line coordinates
A line which makes an angle θ {\displaystyle \theta } with the x-axis, and whose x-intercept is denoted s {\displaystyle s} , is represented by the dual number
z = tan ( θ / 2 ) ( 1 + ε s ) . {\displaystyle z=\tan(\theta /2)(1+\varepsilon s).}
The above doesn't make sense when the line is parallel to the x-axis. In that case, if θ = π {\displaystyle \theta =\pi } then set z = − 2 ε R {\displaystyle z={\frac {-2}{\varepsilon R}}} where R {\displaystyle R} is the y-intercept of the line. This may not appear to be valid, as one is dividing by a zero divisor, but this is a valid point on the projective dual line. If θ = 2 π {\displaystyle \theta =2\pi } then set z = 1 2 ε R {\displaystyle z={\frac {1}{2}}\varepsilon R} . Finally, observe that these coordinates represent oriented lines. An oriented line is an ordinary line with one of two possible orientations attached to it. This can be seen from the fact that if θ {\displaystyle \theta } is increased by π {\displaystyle \pi } then the resulting dual number representative is not the same.
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