Laguerre transformations

Mappings of the form ${\displaystyle z\mapsto {\frac {pz-q}{{\bar {q}}z+{\bar {p}}}}}$ 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 ${\displaystyle z\mapsto -z}$ represents a reflection about the x-axis.

The transformation ${\displaystyle z\mapsto 1/z}$ expresses a reflection about the y-axis.

Observe that if ${\displaystyle U}$ is the matrix representation of any combination of the above three transformations, but normalised so as to have determinant ${\displaystyle 1}$, then ${\displaystyle U}$ satisfies ${\displaystyle UU^{*}=U^{*}U=I}$ where ${\displaystyle U^{*}}$ means ${\displaystyle {\overline {U}}^{\mathrm {T} }}$. 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.

An axial dilation by ${\displaystyle t}$ units is a transformation of the form ${\displaystyle {\frac {z+(\varepsilon t/2)}{(-\varepsilon t/2)z+1}}}$. An axial dilation by ${\displaystyle t}$ units increases the radius of all oriented circles by ${\displaystyle t}$ units while preserving their centres. If a circle has negative orientation, then its radius is considered negative, and therefore for some positive values of ${\displaystyle t}$ 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 ${\displaystyle t}$ units maps any line ${\displaystyle z}$ to a line ${\displaystyle z'}$ such that ${\displaystyle z}$ and ${\displaystyle z'}$ are parallel, and the perpendicular distance between ${\displaystyle z}$ and ${\displaystyle z'}$ is ${\displaystyle t}$. Lines that are parallel but have opposite orientations move in opposite directions.

Figure 2: A grid of lines undergoing ${\displaystyle z\mapsto kz}$ for ${\displaystyle k}$ varying between ${\displaystyle 1}$ and ${\displaystyle 10}$.

Figure 3: Two circles that initially differ only in orientation undergoing the transformation ${\displaystyle z\mapsto kz}$ for ${\displaystyle k}$ varying from ${\displaystyle 1}$ and ${\displaystyle 10}$.

The transformation ${\displaystyle z\mapsto kz}$ for a value of ${\displaystyle k}$ 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).

Putting it all together, a general Laguerre transformation in matrix form can be expressed as ${\displaystyle USV^{*}}$ where ${\displaystyle U}$ and ${\displaystyle V}$ are unitary, and ${\displaystyle S}$ is a matrix either of the form ${\displaystyle {\begin{pmatrix}a&0\\0&b\end{pmatrix}}}$ or ${\displaystyle {\begin{pmatrix}a&-b\varepsilon \\b\varepsilon &a\end{pmatrix}}}$ where ${\displaystyle a}$ and ${\displaystyle b}$ are real numbers. The matrices ${\displaystyle U}$ and ${\displaystyle V}$ express Euclidean isometries. The matrix ${\displaystyle S}$ either represents a transformation of the form ${\displaystyle z\mapsto kz}$ or an axial dilation. The resemblance to Singular Value Decomposition should be clear.[5]

Note: In the event that ${\displaystyle S}$ is an axial dilation, the factor ${\displaystyle V}$ can be set to the identity matrix. This follows from the fact that if ${\displaystyle V}$ is unitary and ${\displaystyle S}$ is an axial dilation, then it can be seen that ${\displaystyle SV={\begin{cases}VS,&\det(V)=+1\\VS^{\mathrm {T} },&\det(V)=-1\end{cases}}}$, where ${\displaystyle S^{\mathrm {T} }}$ denotes the transpose of ${\displaystyle S}$. So ${\displaystyle USV^{*}={\begin{cases}(UV^{*})S,&\det(V)=+1\\(UV^{*})S^{\mathrm {T} },&\det(V)=-1\end{cases}}}$.

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 ${\displaystyle s}$, and makes an angle ${\displaystyle \theta }$ with the equator at the point of intersection, can be represented by the complex number ${\displaystyle \tan(\theta /2)(\cos(s)+i\sin(s))}$. In the case where ${\displaystyle \theta =\pi }$ (where the line is literally the same as the equator, but oriented in the opposite direction as when ${\displaystyle \theta =0}$) the oriented line is represented as ${\displaystyle \infty }$. 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.

An example of a sequence of hyperbolic Laguerre transformations that map a circle to a horocycle to a hypercycle and converge towards a line. This uses the split-complex numbers.

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 ${\displaystyle (a,-b^{-1})}$ because the algebra in question is isomorphic to ${\displaystyle \mathbb {R} \oplus \mathbb {R} }$. (Notice though that as a *-algebra, as opposed to a mere algebra, the split-complex numbers are not decomposable in this way). The terms ${\displaystyle a}$ and ${\displaystyle b}$ in ${\displaystyle (a,-b^{-1})}$ 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 ${\displaystyle \mathbb {RP} ^{1}}$, we need ${\displaystyle a}$ and ${\displaystyle b}$ to belong to the projective line ${\displaystyle \mathbb {RP} ^{1}}$ instead of the affine line ${\displaystyle \mathbb {R} ^{1}}$. Indeed, this hints that ${\displaystyle (\mathbb {R} \oplus \mathbb {R} )\mathbb {P} ^{1}\cong \mathbb {R} \mathbb {P} ^{1}\oplus \mathbb {R} \mathbb {P} ^{1}}$.

The analogue of unitary matrices over the split-complex numbers are the isometries of the hyperbolic plane. This is shown by Yaglom.[1] 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.[6][1]

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.

n-dimensional Laguerre space is isomorphic to n + 1 Minkowski space. To associate a point ${\displaystyle P=(x_{1},x_{2},\dotsc ,x_{n},r)}$ in Minkowski space to an oriented hypersphere, intersect the light cone centred at ${\displaystyle P}$ with the ${\displaystyle t=0}$ hyperplane. The group of Laguerre transformations is isomorphic then to the Poincaré group ${\displaystyle \mathbb {R} ^{n,1}\rtimes \operatorname {O} (n,1)}$. 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 ${\displaystyle \mathbb {R} ^{n,1}\rtimes \operatorname {O} ^{+}(n,1)}$. 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 ${\displaystyle \operatorname {Cl} _{2,0,1}(\mathbb {R} )}$, then analogous Clifford algebraic representations are possible in higher dimensions.

If we embed Minkowski space ${\displaystyle \mathbb {R} ^{n,1}}$ in the projective space ${\displaystyle \mathbb {RP} ^{n+1}}$ 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 can be demonstrated using the homothety centred at ${\displaystyle (0,0)}$ by ${\displaystyle t}$ units. Now consider the action of this transformation on two circles: One simply being the point ${\displaystyle (0,0)}$, and the other being a circle of raidus ${\displaystyle 1}$ centred at ${\displaystyle (0,0)}$. These two circles have a Darboux product equal to ${\displaystyle -1}$. Their images under the homothety have a Darboux product equal to ${\displaystyle -t^{2}}$. This therefore only gives a Laguerre transformation when ${\displaystyle t^{2}=1}$.