Laguerre transformations
[1][2][3][4] When studying these transformations, the dual numbers are often interpreted as representing oriented lines on the plane.
Points on this cylinder are in a natural one-to-one correspondence with oriented lines on the plane.
, is represented by the dual number The above doesn't make sense when the line is parallel to the x-axis.
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.
(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.
The matrix representations of oriented circles (which include points but not lines) are precisely the invertible
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.
express rigid body motions (sometimes called direct Euclidean isometries).
The matrix representations of these transformations span a subalgebra isomorphic to the planar quaternions.
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
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.
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.
(Notice though that as a *-algebra, as opposed to a mere algebra, the split-complex numbers are not decomposable in this way).
The analogue of unitary matrices over the split-complex numbers are the isometries of the hyperbolic plane.
in Minkowski space to an oriented hypersphere, intersect the light cone centred at
These transformations are exactly those which preserve a kind of squared distance between oriented circles called their Darboux product.
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
while keeping the transformation group the same, then the points at infinity are oriented flats.
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.
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 number m corresponds to the signed area of the right triangle with base on the interval [(√2,0), (√2, m √2)].
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.
