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 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 points of the dual number projective line can be defined equivalently in two ways:
Line coordinates.
A line which makes an angle formula_16 with the x-axis, and whose x-intercept is denoted formula_17, is represented by the dual number
 formula_18
The above doesn't make sense when the line is parallel to the x-axis. In that case, if formula_19 then set formula_20 where formula_21 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_22 then set formula_23.
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_16 is increased by formula_25 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_26 where formula_21 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 to the formula_12-axis and non-parallel. The other advantage is that these homogeneous coordinates can be interpreted as vectors, allowing us to multiply them by matrices.
Every Laguerre transformation can be represented as a 2×2 matrix whose entries are dual numbers. The matrix representation of formula_29 is formula_30 (but notice that any non-nilpotent scalar multiple of this matrix represents the same Laguerre transformation). Additionally, as long as the determinant of a 2×2 matrix with dual-number entries is not nilpotent, then it represents a Laguerre transformation.
"(Note that in the above, we represent the homogeneous vector formula_31 as a column vector in the obvious way, instead of as a row vector.)"
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 lines" as well as on oriented circles. An oriented circle is an ordinary circle with an orientation represented by a binary value attached to it, which is either formula_32 or formula_33. The only exception is a circle of radius zero, which has orientation equal to formula_34. A point is defined to be an oriented circle of radius zero. If an oriented circle has orientation equal to formula_32, then the circle is said to be "anti-clockwise" oriented; if it has orientation equal to formula_33 then it is "clockwise" oriented. The radius of an oriented circle is defined to be the radius formula_37 of the underlying unoriented circle multiplied by the orientation.
The image of an oriented circle under a Laguerre transformation is another oriented circle. If two oriented figures – either circles or lines – are tangent to each other then their images under a Laguerre transformation are also tangent. Two oriented circles are defined to be tangent if their underlying circles are tangent and their orientations are equal at the point of contact. Tangency between lines and circles is defined similarly. A Laguerre transformation might map a point to an oriented circle which is no longer a point.
An oriented circle can never be mapped to an oriented line. Likewise, an oriented line can never be mapped to an oriented circle. This is opposite to Möbius geometry, where lines and circles can be mapped to each other, but neither can be mapped to points. Both Möbius geometry and Laguerre geometry are subgeometries of Lie sphere geometry, where points and oriented lines can be mapped to each other, but tangency remains preserved.
The matrix representations of oriented circles (which include points but not lines) are precisely the invertible formula_38 skew-Hermitian dual number matrices. These are all of the form formula_39 (where all the variables are real, and formula_40). The set of oriented lines tangent to an oriented circle is given by formula_41 where formula_42 denotes the projective line over the dual numbers formula_43. Applying a Laguerre transformation represented by formula_44 to the oriented circle represented by formula_45 gives the oriented circle represented by formula_46. The radius of an oriented circle is equal to the half the trace. The orientation is then the sign of the trace.
Profile.
"Note that the animated figures below show some oriented lines, but without any visual indication of a line's orientation (so two lines that differ only in orientation are displayed in the same way); oriented circles are shown as a set of oriented tangent lines, which results in a certain visual effect."
The following can be found in Isaak Yaglom's "Complex numbers in geometry" and a paper by Gutin entitled "Generalizations of singular value decomposition to dual-numbered matrices".
Unitary matrices.
Mappings of the form formula_47 express rigid body motions (sometimes called "direct Euclidean isometries"). The matrix representations of these transformations span a subalgebra isomorphic to the planar quaternions.
The mapping formula_48 represents a reflection about the x-axis.
The transformation formula_49 expresses a reflection about the y-axis.
Observe that if formula_50 is the matrix representation of any combination of the above three transformations, but normalised so as to have determinant formula_32, then formula_50 satisfies formula_53 where formula_54 means formula_55. We will call these "unitary" matrices. Notice though that these are unitary in the sense of the dual numbers and not the complex numbers. The unitary matrices express precisely the Euclidean isometries.
Axial dilation matrices.
An "axial dilation" by formula_56 units is a transformation of the form formula_57. An axial dilation by formula_56 units increases the radius of all oriented circles by formula_56 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_56 the circle actually shrinks. An axial dilation is depicted in Figure 1, in which two circles of opposite orientations undergo the same axial dilation.
On lines, an axial dilation by formula_56 units maps any line formula_3 to a line formula_63 such that formula_3 and formula_63 are parallel, and the perpendicular distance between formula_3 and formula_63 is formula_56. Lines that are parallel but have opposite orientations move in opposite directions.
Real diagonal matrices.
The transformation formula_69 for a value of formula_70 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).
A general decomposition.
Putting it all together, a general Laguerre transformation in matrix form can be expressed as formula_71 where formula_50 and formula_73 are unitary, and formula_74 is a matrix either of the form formula_75 or formula_76 where formula_77 and formula_78 are real numbers. The matrices formula_50 and formula_73 express Euclidean isometries. The matrix formula_74 either represents a transformation of the form formula_69 or an axial dilation. The resemblance to Singular Value Decomposition should be clear.
"Note: In the event that formula_74 is an axial dilation, the factor formula_73 can be set to the identity matrix. This follows from the fact that if formula_73 is unitary and formula_74 is an axial dilation, then it can be seen that formula_87, where formula_88 denotes the transpose of formula_74. So formula_90."
Other number systems and the parallel postulate.
Complex numbers and elliptic geometry.
A question arises: What happens if the role of the dual numbers above is changed to the complex numbers? In that case, the complex numbers represent oriented lines in the elliptic plane (the plane which elliptic geometry takes places over). This is in contrast to the dual numbers, which represent oriented lines in the Euclidean plane. The elliptic plane is essentially a sphere (but where antipodal points are identified), and the lines are thus great circles. We can choose an arbitrary great circle to be the equator. The oriented great circle which intersects the equator at longitude formula_17, and makes an angle formula_16 with the equator at the point of intersection, can be represented by the complex number formula_93. In the case where formula_19 (where the line is literally the same as the equator, but oriented in the opposite direction as when formula_95) the oriented line is represented as formula_96. Similar to the case of the dual numbers, the unitary matrices act as isometries of the elliptic plane. The set of "elliptic Laguerre transformations" (which are the analogues of the Laguerre transformations in this setting) can be decomposed using Singular Value Decomposition of complex matrices, in a similar way to how we decomposed Euclidean Laguerre transformations using an "analogue of Singular Value Decomposition for dual-number matrices".
Split-complex numbers and hyperbolic geometry.
If the role of the dual numbers or complex numbers is changed to the split-complex numbers, then a similar formalism can be developed for representing oriented lines on the hyperbolic plane instead of the Euclidean or elliptic planes: A split-complex number can be written in the form formula_97 because the algebra in question is isomorphic to formula_98. (Notice though that as a *-algebra, as opposed to a mere algebra, the split-complex numbers are not decomposable in this way). The terms formula_77 and formula_78 in formula_97 represent points on the boundary of the hyperbolic plane; they are respectively the starting and ending points of an oriented line. Since the boundary of the hyperbolic plane is homeomorphic to the projective line formula_102, we need formula_77 and formula_78 to belong to the projective line formula_102 instead of the affine line formula_106. Indeed, this hints that formula_107.
The analogue of unitary matrices over the split-complex numbers are the isometries of the hyperbolic plane. This is shown by Yaglom. Furthermore, the set of linear fractional transformations can be decomposed in a way that resembles Singular Value Decomposition, but which also unifies it with the Jordan decomposition.
Summary.
We therefore have a correspondence between the three planar number systems (complex, dual and split-complex numbers) and the three non-Euclidean geometries. The number system that corresponds to Euclidean geometry is the dual numbers.
In higher dimensions.
Euclidean.
n-dimensional Laguerre space is isomorphic to "n" + 1 Minkowski space. To associate a point formula_108 in Minkowski space to an oriented hypersphere, intersect the light cone centred at formula_109 with the formula_110 hyperplane. The group of Laguerre transformations is isomorphic then to the Poincaré group formula_111. These transformations are exactly those which preserve a kind of squared distance between oriented circles called their Darboux product. The "direct Laguerre transformations" are defined as the subgroup formula_112. In 2 dimensions, the direct Laguerre transformations can be represented by 2×2 dual number matrices. If the 2×2 dual number matrices are understood as constituting the Clifford algebra formula_113, then analogous Clifford algebraic representations are possible in higher dimensions.
If we embed Minkowski space formula_114 in the projective space formula_115 while keeping the transformation group the same, then the points at infinity are oriented flats. We call them "flats" because their shape is flat. In 2 dimensions, these are the oriented lines.
As an aside, there are two non-equivalent definitions of a Laguerre transformation: Either as a Lie sphere transformation that preserves oriented flats, or as a Lie sphere transformation that preserves the Darboux product. We use the latter convention in this article. Note that even in 2 dimensions, the former transformation group is more general than the latter: A homothety for example maps oriented lines to oriented lines, but does not in general preserve the Darboux product. This