Laguerre transformations

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 (ignoring possible orientation reversals).
Strictly speaking, these transformations act on the dual number projective line, which adjoins to the dual numbers a line 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 formula_1 where formula_2 are all dual numbers, formula_3 lies on the dual number projective line, and formula_4 is not a zero divisor.
A dual number is a hypercomplex number of the form formula_5 where formula_6 but formula_7. This can be compared to the complex numbers which are of the form formula_8 where formula_9. The "dual number projective line" adjoins to the dual numbers a set of numbers of the form formula_10 for any real formula_11.
Line coordinates.
A line which makes an angle formula_12 with the x-axis, and whose x-intercept is denoted formula_13, is represented by the dual number
The above doesn't make sense when the line is parallel to the x-axis. In that case, if formula_15 then set formula_16 where formula_17 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 formula_18 then set formula_19.
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 formula_12 is increased by formula_21 then the resulting dual number representative is not the same.
Matrix representations.
It's possible to express the above line coordinates as homogeneous coordinates formula_22 where formula_17 is the perpendicular distance of the line from the origin. This representation has numerous advantages: One advantage is that there is no need to break into different cases, such as parallel and non-parallel. The other advantage is that these homogeneous coordinates can be interpreted as vectors, allowing us to multiply them by a matrix.
Every Laguerre transformation can be represented as a 2x2 matrix whose entries are dual numbers. Additionally, as long as the determinant of a 2x2 dual-numbered matrix is not nilpotent, then it represents a Laguerre transformation.
Points, oriented lines and oriented circles.
Laguerre transformations do not act on points. This is because if three oriented lines pass through the same point, their images under a Laguerre transformation do not have to meet at one point.
Laguerre transformations can be seen as acting on oriented circles as well as oriented lines. An oriented circle is one which either has a clockwise or anti-clockwise orientation. An anti-clockwise orientation is considered to be positive, while a clockwise orientation is considered to be negative. The radius of a negatively oriented circle is "negative". Whenever a set of oriented lines are tangent to the same oriented circle, their images under a Laguerre transformation share this property, but possibly for a different circle. An oriented line is tangent to an oriented circle if the two figures touch and their orientations agree.
Profile.
The following can be found in Isaak Yaglom's "Complex numbers in geometry".
Mappings of the form formula_24 express rigid body motions. The matrix representations of these transformations span a subalgebra isomorphic to the dual-complex numbers.
The mapping formula_25 represents a reflection about the x-axis, followed by a reversal of orientation.
The transformation formula_26 expresses a reflection about the y-axis, followed by a reversal of orientation.
An "axial dilatation" by formula_27 units is a transformation of the form formula_28. A dilatation by formula_27 units increases the radius of all oriented circles by formula_27 units while preserving their centres. If a circle has negative orientation, then its radius is considered negative, and therefore for some positive values of formula_27 the circle actually shrinks. A progressive dilatation is depicted in Figure 1, in which two circles of opposite orientations undergo the same dilatation. 
On lines, an axial dilatation by formula_27 units maps any line formula_3 to a line formula_34 such that formula_3 and formula_34 are parallel, and the perpendicular distance between formula_3 and formula_34 is formula_27. Lines that are parallel but have opposite orientations move in opposite directions. 
The transformation formula_40 for a value of formula_11 that's real preserves the x-intercept of a line, while changing its angle to the x-axis. See Figure 2 to observe the effect on a grid of lines (including the x axis in the middle) and Figure 3 to observe the effect on two circles that differ initially only in orientation (to see that the outcome is sensitive to orientation). 
All Laguerre transformations are either:
Conformal interpretation.
In this section, we interpret Laguerre transformations differently from in the rest of the article. When acting on line coordinates, Laguerre transformations are "not" understood to be conformal in the sense described here. This is clearly demonstrated in Figure 2.
The Laguerre transformations preserve angles when the proper angle for the dual number plane is identified. When a ray "y" = "mx", "x" ≥ 0, and the positive x-axis are taken for sides of an angle, the slope "m" is the magnitude of this angle.
This number "m" corresponds to the area of the right triangle with base on the interval [0, √2]. The line {1 + "a"ε: "a" ∈ ℝ}, with the dual number multiplication, forms a subgroup of the unit dual numbers, each element being a shear mapping when acting on the dual number plane. Other angles in the plane are generated by such action, and since shear mapping preserves area, the size of these angles is the same as the original.
Note that the inversion "z" to 1/"z" leaves angle size invariant. As the general Laguerre transformation is generated by translations, dilations, shears, and inversions, and all of these leave angle invariant, the general Laguerre transformation is conformal in the sense of these angles.