Checkerboard incircular nets: Laguerre geometry and parametrisation

Abstract

We present a procedure which allows one to integrate explicitly the class of checkerboard IC-nets which has recently been introduced as a generalisation of incircular (IC) nets. The latter class of privileged congruences of lines in the plane is known to admit a great variety of geometric properties which are also present in the case of checkerboard IC-nets. The parametrisation obtained in this manner is reminiscent of that associated with elliptic billiards. Connections with discrete confocal coordinate systems and the fundamental QRT maps of integrable systems theory are made. The formalism developed in this paper is based on the existence of underlying pencils of conics and quadrics which is exploited in a Laguerre geometric setting.

Appendix: Laguerre geometry

Here, we present the basic facts about Laguerre geometry, focussing on the Blaschke cylinder model employed in this paper for studying checkerboard IC-nets. We begin with the more fundamental Lie sphere geometry. Lie sphere geometry in the plane is the geometry of oriented circles and lines. These are described as elements of the Lie quadric

$$\begin{aligned} {\mathscr {L}}=P(\mathbb {L}^{3,2}), \quad \mathbb {L}^{3,2}=\{x\in \mathbb {R}^{3,2}| <x,x>_{\mathbb {R}^{3,2}}=0\}. \end{aligned}$$

Let $e_1,e_2,e_3,e_4,e_5$ be an orthonormal basis with signature $(+++--)$. For our purposes, another basis $e_1, e_2, e_5, e_\infty , e_0$ defined by

$$\begin{aligned} e_0=\frac{1}{2}(e_4-e_3),\quad e_\infty =\frac{1}{2}(e_4+e_3),\quad <e_0,e_\infty >=-\,\frac{1}{2} \end{aligned}$$

turns out to be more convenient. Elements of $\mathscr {L}$ with non-vanishing $e_0$-component are identified with oriented circles $|{\varvec{x}}-{\varvec{c}}|^2=r^2$, centred at ${\varvec{c}}\in \mathbb {R}^2$ and of radius $r\in \mathbb {R}$:

$$\begin{aligned} s={\varvec{c}}+re_5+(|{\varvec{c}}|^2-r^2)e_\infty +e_0. \end{aligned}$$

(42)

Points are circles of radius $r=0$, and oriented lines $({\varvec{v}},{\varvec{x}})_{\mathbb {R}^2}=d$ are elements of $\mathscr {L}$ with vanishing $e_0$-component:

$$\begin{aligned} p={\varvec{v}}+e_5+2de_\infty . \end{aligned}$$

(43)

The incidence $<p,s>=0$ is the condition

$$\begin{aligned} ({\varvec{c}},{\varvec{v}})-r=d \end{aligned}$$

(44)

of oriented contact of a circle and a line.

The Lie sphere transformation group $\textit{PO}(3,2)$ acting on $P(\mathbb {R}^{3,2})$ preserves the Lie quadric $\mathscr {L}$ and maps oriented circles and lines to oriented circles and lines, preserving oriented contact. Its subgroup of Laguerre transformations preserves the set of straight lines or, equivalently, the hyperplane

$$\begin{aligned} {\mathsf P}=\mathrm{span}\{e_1,e_2,e_5,e_\infty \}=\{w\in \mathbb {R}^{3,2}|<w,e_\infty >=0\}. \end{aligned}$$

Direct computation shows that the elements of $\textit{PO}(3,2)$ preserving the hyperplane $\mathsf P$ are of the form

$$\begin{aligned} \left( \begin{matrix} \lambda B &{}\quad 0 &{}\quad \alpha \\ b^T &{}\quad 1 &{}\quad \nu \\ 0 &{}\quad 0 &{}\quad \lambda ^{-2} \end{matrix}\right) \end{aligned}$$

in the basis $e_1,e_2,e_5,e_\infty ,e_0$, where

$$\begin{aligned} B\in O(2,1),\quad b\in \mathbb {R}^{2,1},\quad \alpha =\frac{\lambda }{2}Bb,\quad \nu =\frac{\lambda }{4}(b,b)_{\mathbb {R}^{2,1}},\quad \lambda \in \mathbb {R}. \end{aligned}$$

In order to pass to the Blaschke cylinder model of Laguerre geometry, we confine ourselves to the subspace ${\mathscr {P}}=\mathrm{span}\{e_1,e_2,e_5,e_\infty \}$. Elements of this space can be identified with straight lines, described (projectively) by

$$\begin{aligned} \tilde{p}=\tilde{{\varvec{v}}}+\tilde{s}e_5+2\tilde{d}e_\infty \end{aligned}$$

as points of the Blaschke cylinder

$$\begin{aligned} {\mathscr {Z}}=\{ [\tilde{p}]\in P(\mathbb {R}^{2,1,1})| |\tilde{{\varvec{v}}}|^2=\tilde{s}^2\}. \end{aligned}$$

(45)

Identification with (43) is made via the normalisation of the $e_0$-component: ${\varvec{v}}=\tilde{{\varvec{v}}}/\tilde{s}$, $d=\tilde{d}/\tilde{s}$. It is noted that the symmetry with the description of circles (42) in Lie sphere geometry is no longer present in the Blaschke cylinder model, and oriented circles are described as the sets of all straight lines in oriented contact, i.e., the sets of lines satisfying the condition (44), that is

$$\begin{aligned} S= \{({\varvec{v}},d)\in \mathbb {R}^3| ({\varvec{c}},{\varvec{v}})_{\mathbb {R}^2}-r=d \}. \end{aligned}$$

Furthermore, Laguerre transformations restricted to the subspace of lines ${\mathsf P}=\mathrm{span}\{e_1,e_2,e_5,e_\infty \}$ are of the form

$$\begin{aligned} A= \left( \begin{matrix} \lambda B &{}\quad 0 \\ b^T &{}\quad 1 \end{matrix}\right) . \end{aligned}$$

(46)

Theorem 12

The group of Laguerre transformations in the Blaschke (projective) cylinder model (in the basis $e_1,e_2,e_5,e_\infty$) is represented by matrices of the form (46), where $B\in O(2,1),\, b\in \mathbb {R}^{2,1},\, \lambda \in \mathbb {R}$. These transformations preserve the Blaschke cylinder (45).

We conclude by observing that Euclidean motions

$$\begin{aligned} {\varvec{x}}\rightarrow \tilde{{\varvec{x}}}=R{\varvec{x}}+\varDelta , \quad R\in O(2),\quad \varDelta \in \mathbb {R}^2 \end{aligned}$$

are particular Laguerre transformations, the corresponding matrix of which is given by (46) with

$$\begin{aligned} B=\left( \begin{matrix} R^T &{}\quad 0 \\ 0 &{}\quad 1 \end{matrix}\right) ,\quad b=\left( \begin{matrix} 2R\varDelta \\ 0 \end{matrix}\right) \end{aligned}$$

(47)

and $\lambda =1$.