Ma–Schlenker c-Octahedra in the 2-Sphere

Abstract

We present constructions inspired by the Ma–Schlenker example of “Non-rigidity of spherical inversive distance circle packings” (Discrete Comput Geom 47(3):610–617, 2012). In contrast to the use in Ma and Schlenker (2012) of an infinitesimally flexible Euclidean polyhedron, embeddings in de Sitter space, and Pogorelov maps, our elementary constructions use only the inversive geometry of the 2-sphere.

1 Introduction

In [2], Bowers and Stephenson questioned whether inversive distance circle packings of surfaces are uniquely determined by their underlying triangulation and the inversive distances between the pairs of adjacent circles. Guo [3] confirmed the local rigidity of inversive distance circle packings on closed orientable surfaces of positive genus, and subsequently Luo [4] verified the global rigidity of these packings, answering the Bowers–Stephenson question in the affirmative. Contrasted to this is the beautiful and surprising example of Ma and Schlenker in [5] that provides a counterexample to uniqueness in the spherical case. They produced pairs of packing radii that determine pairs of geodesic triangulations and circle packings on the 2-sphere ${\mathbb {S}}^{2}$ realizing the same inversive distance data, but for which there is no inversive transformation taking one of the circle patterns to the other. This was doubly surprising as the famous Koebe–Andre’ev–Thurston Circle Packing Theorem implies uniqueness of spherical packings up to inversive equivalence whenever the edge labels are all in the unit interval—the case of tangent or overlapping circle packings.

The ingredients of Ma and Schlenker’s example are Schönhardt’s twisted octahedron (an infinitesimally flexible polyhedron in Euclidean space ${\mathbb {E}}^{3}$), embeddings in de Sitter space ${\mathbb {S}}^{3}_{1}$, and special properties of the Pogorelov map between different geometries. In contrast, we provide a construction of a large family of Ma–Schlenker-like examples using only inversive geometry. In fact, we show how to construct many counterexamples to the uniqueness of inversive distance circle packings in the 2-sphere.

2 Preliminaries from Circle Packing and Inversive Geometry

2.1 Inversive Distance in the Plane and the 2-Sphere

Let $C_{1}$ and $C_{2}$ be distinct circles in the complex plane ${\mathbb {C}}$ centered at the respective points $p_{1}$ and $p_{2}$, of respective radii $r_{1}$ and $r_{2}$, and bounding the respective companion disks $D_{1}$ and $D_{2}$. The inversive distance $\langle C_{1}, C_{2} \rangle$ between $C_{1}$ and $C_{2}$ is

$$\begin{aligned} \langle C_{1}, C_{2} \rangle = \frac{|p_{1}-p_{2}|^{2} - r_{1}^{2} - r_{2}^{2}}{2r_{1}r_{2}}. \end{aligned}$$

(1.1)

The absolute inversive distance between distinct circles is the absolute value of the inversive distance and is a Möbius invariant of the placement of two circles in the plane. The important geometric facts that make the inversive distance useful in inversive geometry and circle packing are as follows. When $\langle C_{1},C_{2}\rangle > 1$, $D_{1} \cap D_{2} = \emptyset$ and $\langle C_{1},C_{2}\rangle = \cosh \delta$, where $\delta$ is the hyperbolic distance between the totally geodesic hyperbolic planes in the upper-half-space model ${\mathbb {C}} \times (0,\infty )$ of ${\mathbb {H}}^{3}$ whose ideal boundaries are $C_{1}$ and $C_{2}$. When $\langle C_{1},C_{2}\rangle = 1$, $D_{1}$ and $D_{2}$ are tangent at their single point of intersection. When $1 >\langle C_{1},C_{2}\rangle \ge 0$, $D_{1}$ and $D_{2}$ overlap with angle $0 < \theta \le \pi /2$ with $\langle C_{1},C_{2}\rangle = \cos \theta$. In particular, $\langle C_{1},C_{2}\rangle = 0$ precisely when $\theta = \pi /2$. We shall have no need to use the case when $\langle C_{1},C_{2}\rangle < 0$, when $D_{1}$ and $D_{2}$ overlap by an angle greater than $\pi /2$.

In the 2-sphere ${\mathbb {S}}^{2}$, the inversive distance may be expressed as¹

$$\begin{aligned} \langle C_{1}, C_{2} \rangle = \frac{-\cos \sphericalangle ( p_{1},p_{2} ) + \cos (r_{1}) \cos (r_{2})}{\sin (r_{1}) \sin (r_{2})}. \end{aligned}$$

(1.2)

Here, $\sphericalangle (p_{1},p_{2})$ denotes the spherical distance between the centers, $p_{1}$ and $p_{2}$, of the respective circles² $C_{1}$ and $C_{2}$ with respective spherical radii $r_{1}$ and $r_{2}$.³ Stereographic projection to the plane ${\mathbb {C}}$ preserves the absolute inversive distance of circle pairs and, as long as neither of the two companion disks $D_{1}$ and $D_{2}$ contains the north pole, it preserves the inversive distance.

2.2 Edge-Labeled Triangulations of the 2-Sphere and Inversive Distance Circle Packings

We are concerned here with configurations of circles in the 2-sphere ${\mathbb {S}}^{2}$ with a specified pattern of inversive distances. Let K be an abstract oriented triangulation of ${\mathbb {S}}^{2}$ and $\beta :E(K) \rightarrow {\mathbb {R}}$ a mapping defined on E(K), the set of edges of K. Call K together with $\beta$ an edge-labeled triangulation with edge label $\beta$ and denote it as $K_{\beta }$. We denote an edge of K with vertices u and v by uv, and an oriented face with vertices u, v, and w ordered respecting the orientation of K by uvw.

Definition 1.1

A circle realization for $K_{\beta }$ is a collection ${\mathscr {C}} = \{C_{v} : v \in V(K)\}$ of circles in either the plane ${\mathbb {C}}$ or the 2-sphere ${\mathbb {S}}^{2}$ indexed by the vertex set V(K) of K such that $\langle C_{u}, C_{v}\rangle = \beta (uv )$ whenever uv is an edge of K. When uv is an edge of K, the corresponding circles $C_{u}$ and $C_{v}$ are said to be adjacent.

An inversive distance circle packing for $K_{\beta }$, or circle packing for short, is a circle realization in the 2-sphere where the circles are placed in a way that the circles $C_{u}$, $C_{v}$, and $C_{w}$ form a positively oriented triple in ${\mathbb {S}}^{2}$ whenever uvw is a positively oriented face of K. We will confine our interest to a restricted class of circle packings, those circle packings that are produced by the use of polyhedral metrics, as presented by Ma and Schlenker in their construction. These are precisely those circle realizations that produce isomorphic copies of the triangulation K by connecting the centers of adjacent circles by geodesic segments. We warn the reader, though, that when $\beta$ may take values greater than unity so that adjacent circles are separated, this property of a geodesic triangulation in the pattern of K appearing when connecting centers in general is not preserved by Möbius transformations.

2.3 Möbius Flows

The most important ingredients from inversive geometry needed to construct our examples are the Möbius flows associated to two distinct circles in the extended complex plane. Here are the pertinent facts. Circles $C_{1} \ne C_{2}$ lie in a unique coaxial family ${\mathcal {A}}_{C_{1},C_{2}}$ of circles in the extended complex plane $\widehat{\mathbb {C}}$ whose elements serve as the flow lines of certain 1-parameter subgroups of the Möbius group Möb$(\widehat{\mathbb {C}})\cong {\mathrm {PSL}}(2, {\mathbb {C}})$ acting on the extended plane as linear fractional transformations. The family ${\mathcal {A}}_{C_{1},C_{2}}$ is invariant under these flows. When $C_{1}$ and $C_{2}$ are disjoint, any such flow is an elliptic flow conjugate in Möb$(\widehat{\mathbb {C}})$ to a standard rotation flow of the form $t \mapsto R_{\lambda t}$, where $\lambda \ne 0$ and $R_{\lambda t}$ is the rotation $z \mapsto e^{\lambda \varvec{i} t} z$. When $C_{1}$ and $C_{2}$ meet in a single point p, the flow is parabolic and is conjugate in Möb$(\widehat{\mathbb {C}})$ to a standard translation flow of the form $t\mapsto T_{\lambda t}$, where $\lambda \ne 0$ and $T_{\lambda t}$ is the translation $z \mapsto z + \lambda t$. Finally, when $C_{1}$ and $C_{2}$ meet in two distinct points a and b, the flow is hyperbolic and is conjugate in Möb$(\widehat{\mathbb {C}})$ to a standard scaling flow of the form $t \mapsto S_{\lambda t}$, where $\lambda \ne 0$ and $S_{\lambda t}$ is the scaling map $z \mapsto e^{\lambda t} z$. All the flows determined by two fixed circles $C_{1}$ and $C_{2}$ are said to be equivalent flows, with two such flows differing only in the value of the parameter $\lambda$, with $|\lambda |$ the speed of the flow. Note that any two distinct circles in the coaxial family ${\mathcal {A}}_{C_{1},C_{2}}$ determine the same Möbius flows as $C_{1}$ and $C_{2}$. Of course, any Möbius flow preserves the inversive distances between circles.

Each coaxial family ${\mathcal {A}}$ has an associated orthogonal complement ${\mathcal {A}}^{\perp }$, this also a coaxial family for which each circle of ${\mathcal {A}}$ is orthogonal to each circle of ${\mathcal {A}}^{\perp }$. In fact, ${\mathcal {A}}^{\perp }$ is exactly the collection of circles (and lines) that are orthogonal to every member of ${\mathcal {A}}$, and of course ${\mathcal {A}}^{\perp \perp } = {\mathcal {A}}$. Any flow whose flow lines are the circles of ${\mathcal {A}}$ is generated by inversions through the circles of the orthogonal complement ${\mathcal {A}}^{\perp }$.

3 Ma–Schlenker c-Octahedra: The Examples Writ Large

In this section we detail examples of pairs of spherical circle packings for a fixed edge-labeled triangulation that fail to be inversive equivalent. These were discovered by studying the properties of the first family of such examples constructed by Ma and Schlenker in [5] using fairly sophisticated geometric constructions. Ours are constructed using Möbius flows and, having properties reminiscent of the Ma–Schlenker examples, will be named after them. This section presents a general construction of such examples and describes their important properties, and the next verifies the claimed properties.

Let ${\mathscr {O}}$ be the octahedral triangulation of the 2-sphere with six vertices, each of valence four, twelve edges, and eight faces, and combinatorially equivalent to the boundary of a regular octahedron. The 1-skeleton graph of ${\mathscr {O}}$ is shown embedded in the plane in Fig. 1(a).

Definition 2.1

The edge-labeled triangulation ${\mathscr {O}}_{\beta }$ is called a Ma–Schlenker octahedron provided $\beta$ takes a constant value $a\ge 0$ on the edges of a fixed face uvw, a constant value $d\ge 0$ on the edges of its opposite face $w'v'u'$, and alternates between the values $b\ge 0$ and $c\ge 0$ on the ‘teepee’ of edges connecting the vertices of these two opposite faces, as in Fig. 2. When we need to emphasize the values of a, b, c, and d, we denote ${\mathscr {O}}_{\beta }$ as ${\mathscr {O}}(a,b,c,d)$.

Fig. 1

A critical Ma–Schlenker circle octahedron and two non-inversive equivalent nearby patterns with the same inversive distances on the graph. (a) Octahedral graph labeled with the inversive distances from (b). (b) A planar circle pattern realizing the octahedral graph in (a). (c) Two non-equivalent realizations with inversive distance 37 on the outer edges

Fig. 2

The typical Ma–Schlenker octahedron ${\mathscr {O}}(a,b,c,d)$

Definition 2.2

A circle packing for a Ma–Schlenker octahedron is called a Ma–Schlenker circle-octahedron, or a Ma–Schlenker c-octahedron for short. A pair of circle packings, ${\mathscr {C}}$ and ${\mathscr {C}}'$, for the same Ma–Schlenker octahedron ${\mathscr {O}}_{\beta }$ is called a Ma–Schlenker pair provided ${\mathscr {C}}$ and ${\mathscr {C}}'$ are not inversive equivalent.

Fig. 3

The construction for a particular Ma–Schlenker pair. (a) Start with three circles of equal radii centered at the vertices of an equilateral triangle. Extend rays in order around the edges. (b) Take three circles of equal radii at a fixed distance along each ray to form the outer face. (c) Flow the outer circles symmetrically until the inversive distance reaches a minimum value. This is the critical circle realization. (d) Flow to find a pair of circle realizations (solid and dashed) near the critical packing that have equal inversive distances on the outer face. These are not Möbius equivalent

3.1 The Construction

We now describe a method for constructing Ma–Schlenker pairs. It begins with a construction of a 1-parameter family of planar circle realizations of Ma–Schlenker octahedra ${\mathscr {O}}(a,b,c,d(t))$ where a, b, and c are fixed while $d = d(t)$ varies. Figure 3 presents a graphical description of this construction in a special case. Fix a non-negative number a and choose three circles $C_{u}$, $C_{v}$, and $C_{w}$ in the complex plane that are centered at the respective vertices of an equilateral triangle $\Delta$ so that $\langle C_{u}, C_{v}\rangle = \langle C_{v}, C_{w}\rangle = \langle C_{w}, C_{u}\rangle =a$. Normalize by insisting that $\Delta$ have side length 2 with circle $C_{u}$ centered on $z=-1$, $C_{v}$ centered on $z=1$, and $C_{w}$ centered on $z=\varvec{i}\sqrt{3}$, placing the incenter of $\Delta$ at $z = \varvec{i}/\sqrt{3}$. Let ${\mathcal {A}} = {\mathcal {A}}_{C_{u},C_{v}}$ be the coaxial family containing $C_{u}$ and $C_{v}$ and note that the line of centers of ${\mathcal {A}}$ is the real axis ${\mathbb {R}}$ and the radical axis of ${\mathcal {A}}$ is the imaginary axis ${\mathbb {R}}\varvec{i}$. Choose an initial circle C centered on the positive real axis such that $c = \langle C_{v} , C \rangle \ge 0$ and let $b =\langle C_{u}, C\rangle$. Note that $b > c$. Let $\mu = \{\mu _{t}: t\in {\mathbb {R}}\}$ be the unit speed Möbius flow determined by ${\mathcal {A}}$ that is counterclockwise on the circle $C_{v}$, and let $1< x_{1} < x_{2}$ be the points of intersection of C with the real axis. Let $A_{1}, A_{2} \in {\mathcal {A}}$ be the circles in the family ${\mathcal {A}}$ containing $x_{1}$ and $x_{2}$, respectively. For any real number t, $\mu _{t} (C)$ is a circle tangent to both $A_{1}$ and $A_{2}$ and sits between them, external to the disk bounded by $A_{1}$ and internal to the disk bounded by $A_{2}$. See Fig. 4 for illustrations of the family $\{\mu _{t}(C) : t\in {\mathbb {R}}\}$ when the flow is hyperbolic ($0 \le a <1$), parabolic ($a = 1$), and elliptic ($a>1$). Since the flow $\mu$ preserves the individual circles of the coaxial system ${\mathcal {A}}$, $\langle C_{u}, \mu _{t} (C) \rangle = b$ and $\langle C_{v} , \mu _{t}(C) \rangle = c$ for all $t\in {\mathbb {R}}$.

Fig. 4

Möbius flows from the construction. The smaller gray circle is $A_1$ and the larger is $A_2$. (a) Hyperbolic ($0 \le a < 1$). (b) Parabolic ($a = 1$). (c) Elliptic ($a > 1$)

Let ${\mathfrak {r}}$ be the counterclockwise rotation of the plane ${\mathbb {C}}$ through angle $2\pi /3$ with fixed point $\varvec{i}/\sqrt{3}$, the incenter of the triangle $\Delta$. The circle configuration ${\mathscr {C}}(t)$ is defined as the collection

$$\begin{aligned} {\mathscr {C}}(t) = \big \{ C_{u}, C_{v}, C_{w}, C_{w'}(t) = \mu _{t}(C), C_{u'}(t) = {\mathfrak {r}}(\mu _{t}(C)), C_{v'}(t) = {\mathfrak {r}}^{2}(\mu _{t}(C)) \big \}.\nonumber \\ \end{aligned}$$

(2.1)

The properties of ${\mathscr {C}}(t)$ of interest to us are, first, for all $t\in {\mathbb {R}}$ the configuration ${\mathscr {C}}(t)$ of circles in the plane is a circle realization for the Ma–Schlenker octahedron ${\mathscr {O}}(a,b,c,d(t))$, where

$$\begin{aligned} d(t) = \langle \mu _{t}(C), {\mathfrak {r}}(\mu _{t}(C)) \rangle , \end{aligned}$$

(2.2)

and second, for all $t\in {\mathbb {R}}$, ${\mathscr {C}}(t)$ has order three rotational symmetry about the incenter of $\Delta$ via the rotation ${\mathfrak {r}}$. Stereographic projection of ${\mathscr {C}}(t)$ to ${\mathbb {S}}^{2}$ defines a 1-parameter family of circle realizations for ${\mathscr {O}}(a,b,c,d(t))$ in the 2-sphere, though these are not a priori circle packings. These realizations possibly must be repositioned to a normalized position by applying Möbius transformations of ${\mathbb {S}}^{2}$ to find appropriate intervals of t-values that produce the desired circle packings.

Fig. 5

The center of the critical circle must fall within the shaded region

Our goal is to find values of the inversive distance parameters $a > 1/2$ and $b > c \ge 0$ so that the function d(t) has a critical value at a parameter value $t= \tau$, with $d(\tau )$ a local minimum for d, and for which the circle $\mu _{\tau }(C)$ is centered in the interior of the closed half-plane ${\mathsf {H}}$ containing $z = 1$, the center of $C_{v}$, and bordered by the line through $z=-1$, the center of $C_{u}$, and the incenter of $\Delta$; see Fig. 5. Moreover, $\mu _{\tau }(C)$ should have positive inversive distance to the unique circle O orthogonal to $C_{u}$, $C_{v}$, and $C_{w}$. The circle O exists since $a >1/2$. The claim is that with these conditions satisfied, Ma–Schlenker pairs may be produced.

To see this, let $O'$ be the unique circle orthogonal to $C_{u'}(\tau )$, $C_{v'}(\tau )$, and $C_{w'}(\tau )$, which exists since $d(\tau ) > 1/2$.⁴ By the order three rotational symmetry of ${\mathscr {C}}(t)$, O and $O'$ are concentric, both centered at $\varvec{i}/\sqrt{3}$. Let $\widetilde{\mathscr {C}}(t)$ be the image of ${\mathscr {C}}(t)$ under stereographic projection to the 2-sphere followed by a Möbius transformation so that (1) the respective images of the circles O and $O'$ are latitudinal circles, the first centered on the south pole and contained in the southern hemisphere, and the second on the north pole and contained in the northern hemisphere, and (2) the circles L and $L'$ have the same radius, L centered on the south pole and $L'$ on the north, where L is the latitudinal circle containing the centers of the projections of $C_{u}$, $C_{v}$, and $C_{w}$, and $L'$ is the latitudinal circle containing the centers of the projections of $C_{u'}(\tau )$, $C_{v'}(\tau )$, and $C_{w'}(\tau )$. Note that the order three rotational symmetry of ${\mathscr {C}}(t)$ translates to an order three rotational symmetry of $\widetilde{\mathscr {C}}(t)$ by a rotation of ${\mathbb {S}}^{2}$ about the axis through the north and south poles.

Our claim is that there is an open interval J of t-values containing $\tau$ for which each $\widetilde{\mathscr {C}}(t)$, for $t\in J$, is a circle packing. Since then the Ma–Schlenker c-octahedron $\widetilde{\mathscr {C}}(\tau )$ provides a minimum value for d(t), variation of t about $\tau$ produces (possibly with further restrictions on a, b, and c) pairs $t < \tau$ and $t' > \tau$ in J for which $\widetilde{\mathscr {C}}(t)$ and $\widetilde{\mathscr {C}}(t')$ form a Ma–Schlenker pair, two circle packings for ${\mathscr {O}}(a,b,c,d)$, where $d= d(t) = d(t')$, that fail to be inversive equivalent.

Definition 2.3

The circle realization ${\mathscr {C}}(\tau )$ is said to be critical, the circle packing $\widetilde{\mathscr {C}}(\tau )$ is a critical Ma–Schlenker c-octahedron, and the corresponding edge-labeled triangulation ${\mathscr {O}}(a,b,c,d(\tau ))$ is a critical Ma–Schlenker octahedron. See Fig. 6 for a 3D visualization of a critical Ma–Schlenker c-octahedron.

Fig. 6

A 3D visualization of a critical Ma–Schlenker octahedron showing the triangulation. (a) A view from the south pole. (b) A view from the north pole. (c) A view from slightly north of the equator. (d) A slight rotation from (c)

The claim of the preceding paragraph, that there is an open interval J of t-values containing $\tau$ for which each $\widetilde{\mathscr {C}}(t)$, for $t\in J$, is a circle packing follows from the fact that the circle $\mu _{\tau }(C)$ is centered in the interior of the closed half-plane ${\mathsf {H}}$. The point is that three points u, v, and w equally spaced on L and $u'$, $v'$ and $w'$ equally spaced on $L'$ with u and $w'$ on the same meridian, v and $u'$ on the same meridian, and w and $v'$ on the same meridian, cut out an octahedral triangulation of ${\mathbb {S}}^{2}$ when points are connected according to the Ma–Schlenker octahedral pattern. Assuming the points are ordered counterclockwise when viewed from the north pole, if now the points along $L'$ are rotated counterclockwise by an angle strictly between 0 and $\pi$, then this triangulation “stretches” to form a triangulation with vertices u, v, and w, and the rotated $u'$, $v'$, and $w'$. It is only when the rotation reaches $\pi$ radians that there is ambiguity as then u and $w'$ are antipodal, and then as $w'$ is rotated past $\pi$ radians, the geodesic arc connecting u to $w'$ moves to the other side of the north pole, and the triangle $uvw'$ now contains the north pole, as do $vwu'$ and $wuv'$. At this point we have lost the triangulation. The point of the half-plane is that the center of $\mu _{\tau }(C)$ in the interior of ${\mathsf {H}}$ guarantees that the position of $w'$ is obtained in this manner by a positive counterclockwise rotation strictly less than $\pi$.

We have used the restriction $a > 1/2$ to ensure the existence of a circle mutually orthogonal to the three circles $C_{u}$, $C_{v}$, and $C_{w}$. This in turn is used to ensure that stereographic projection followed by an appropriate Möbius transformation produces, not just a realization, but a circle packing for a Ma–Schlenker octahedron. This is done for convenience of argument and we mention that even if $a \le 1/2$, when there is no mutually orthogonal circle to $C_{u}$, $C_{v}$, and $C_{w}$, Ma–Schlenker circle packings are still possible.

Fig. 7

A parabolic flow. The parameters are $x_1 = 1.7$, $x_2 = 3$. The highlighted minimum and (local) maximum occur at $\tau = t_\mathrm{min} = 0.121766$ (orange) and $m = t_\mathrm{max} = 0.866025$ (green), and the inversive distances are $d(t_\mathrm{min}) = 18.6065$ and $d(t_\mathrm{max}) = 28.051$ (computed values are rounded). (a) Parabolic flow. (b) Graph of the inversive distance function d(t)

Fig. 8

Hyperbolic flow. The parameters are $x_1 = 2.11803$, $x_2 = 4.06155$, and $y = 0.5$. The highlighted minimum (orange) occurs at $\tau = t_\mathrm{min} = 0.06782$ and the local maximum (green) at $m = t_\mathrm{max} = 1.31696$. The inversive distances are $d(t_\mathrm{min}) = 14.1647$ and $d(t_\mathrm{max}) = 46.8136$. (a) Hyperbolic flow. (b) Detail of the dotted box. (c) Graph of the inversive distance

Fig. 9

An elliptic flow. The parameters are $x_1 = 2$, $x_2 = 6$, and $y = 0.5$. The highlighted minimum (orange) occurs at $\tau = t_\mathrm{min} = 0.0506$, the local maximum (blue) at $m = t_\mathrm{max}^{(1)} = 0.7137$, and the absolute maximum (green) at $M = t_\mathrm{max}^{(2)} = 3.85532$. The inversive distances are $d(t_\mathrm{min}) = 5.000$, $d(t_\mathrm{max}^{(1)}) = 10.4245$, and $d(t_\mathrm{max}^{(2)}) = 391.247$. (a) Elliptic flow. (b) Detail of the dotted box. (c) Graph of the inversive distance, which is periodic of period $2\pi$

4 Ma–Schlenker c-Octahedra: The Devil of the Details

Figures 7, 8 and 9 show examples of the graphs of d(t) for various values of a and choices of initial circle C. The Ma–Schlenker pairs arise from pairs t and $t'$ near $\tau$ and at the same horizontal level on the graph of d(t). A precise description of the derivation of these graphs is given subsequently, after we make some observations.

4.1 The Parabolic Case

Here $a=1$, the circles $C_{u}$, $C_{v}$ and $C_{w}$ are mutually tangent, and the flow is parabolic. An example is shown in Fig. 7(a), where the initial circle C meets the real axis at $x_{1} = 1.7$ and $x_{2} = 3$, with $b = \langle C_{u}, C\rangle = 7.538$ and $c = \langle C_{v}, C\rangle = 0.308$. The three circles $C_{u}$, $C_{v}$ and $C_{w}$ are the dotted circles. The critical circle $C(\tau )$, in orange, gives a minimum value for d(t) and is centered in the half-plane ${\mathsf {H}}$. The graph of d(t) is shown in Fig. 7(b), whose shape is typical of all the examples with $a=1$. The general characteristics of the parabolic flow are the same as for the hyperbolic, which is described in detail in the next paragraph.

4.2 The Hyperbolic Case

Here $1/2 < a\le 1$ and the pairs of circles among $C_{u}$, $C_{v}$, and $C_{w}$ meet at angle $\pi /3 \ge \theta = \cos ^{-1} a \ge 0$. The center of the unique circle O orthogonal to $C_{u}$, $C_{v}$ and $C_{w}$ is the incenter $\varvec{i}/\sqrt{3}$ of $\Delta$, and lies in the bounded interstice formed by $C_{u}$, $C_{v}$ and $C_{w}$. As $t\rightarrow \pm \infty$, the inversive distance parameter $d(t) \rightarrow + \infty$. The absolute minimum value of d(t) occurs at two distinct t-values $0< \tau < \tau '$, which implies that both circles, $C(\tau ) =\mu _{\tau }(C)$ and $C(\tau ') = \mu _{\tau '}(C)$, are centered in the first quadrant of the complex plane. Between these lies a parameter value m at which d(t) obtains a local maximum. The circle $C(m) = \mu _{m}(C)$ is orthogonal to O and inversion $I_{O}$ through the circle O preserves C(m) and exchanges circles $C(\tau )$ and $C(\tau ')$. The center of $C(\tau )$ lies outside of O, and that of $C(\tau ')$ inside. More generally, the inversion $I_{O}$ generates a symmetry of the graph in the following way. Since O is in the orthogonal complement ${\mathcal {A}}_{C_{u},C_{v}}^{\perp }$, the inversion $I_{O}$ preserves the family ${\mathcal {A}} ={\mathcal {A}}_{C_{u},C_{v}}$, merely inverting each circle of ${\mathcal {A}}$ to itself. In particular, since $A_{1}$ and $A_{2}$ are members of ${\mathcal {A}}$ and form the envelope of the family ${\mathcal {B}} = \{\mu _{t}(C) : t \in {\mathbb {R}}\}$ of circles generated by the Möbius flow $\mu$, $I_{O}(A_{1}) = A_{1}$ and $I_{O}(A_{2}) = A_{2}$ form the envelope of the circles of the family $I_{O}({\mathcal {B}})$, implying that $I_{O}$ preserves the family ${\mathcal {B}}$. It follows that $I_{O}$ generates an involution of ${\mathbb {R}}$. Indeed, for each $t\in {\mathbb {R}}$, let $t'$ be the real value for which $\mu _{t'}(C) = I_{O}(\mu _{t}(C))$. Then $t'' =t$ with fixed point $m = m'$. This generates a symmetry of the graph of d(t) with $d(t) = d(t')$ since

$$\begin{aligned} d(t)&= \langle \mu _{t}(C), {\mathfrak {r}}(\mu _{t}(C)) \rangle \\&= \langle I_{O}(\mu _{t}(C)), I_{O}({\mathfrak {r}}(\mu _{t}(C)) )\rangle \\&= \langle I_{O}(\mu _{t}(C)), {\mathfrak {r}}(I_{O}(\mu _{t}(C))) \rangle \\&= \langle \mu _{t'}(C), {\mathfrak {r}}(\mu _{t'}(C)) \rangle =d(t'), \end{aligned}$$

since the inversion $I_{O}$ commutes with the rotation ${\mathfrak {r}}$, as both are centered at $\varvec{i}/\sqrt{3}$.

Figure 8 presents an example where $a= 0.6$ with the angle $\theta$ of intersection of $C_{u}$ and $C_{v}$ equal to $2 \tan ^{-1} (1/2)\cong 0.9273$. The three circles $C_{u}$, $C_{v}$ and $C_{w}$ are suppressed in the figure, and the respective values of b and c are $b=6.689$ and $c=1$. The circles $C_{u}$ and $C_{v}$ meet at $\pm y \varvec{i}$ where $y = 0.5$. In the close up of Fig. 8(b), the three local maximum circles (green) and the six enveloping circles are orthogonal to O (not shown).

4.3 The Elliptic Case

Here $a > 1$ and the facts are similar to those of the hyperbolic case. The primary difference is that d(t) now is periodic of period $\omega = 2\pi / |\lambda |$, where $|\lambda |$ is the speed of the elliptic flow. There are parameter values $0< \tau< m< \tau '< M < \omega$ where $d(\tau ) = d(\tau ')$ is the absolute minimum value, d(m) is a local maximum value, and d(M) is the absolute maximum value. The circles $C(m) = \mu _{m}(C)$ and $C(M) = \mu _{M}(C)$ are orthogonal to O and $C(\tau ) =\mu _{\tau }(C)$ and $C(\tau ')= \mu _{\tau '}(C)$ are centered in the first quadrant and are exchanged by the inversion $I_{O}$, which generates a periodic symmetry of the graph similar to the case articulated in the preceding paragraph.

Figure 9 presents an example where $a= 5/3$ and the coaxial family ${\mathcal {A}}$ has foci $\pm y$, where $y = 0.5$. The three circles $C_{u}$, $C_{v}$ and $C_{w}$ are suppressed in the figure, and the respective values of b and c are $b = 5.846$ and $c=1.227$. In the close up of Fig. 9(b), the three local maximum circles (blue), the three absolute maximum circles (green), and the six enveloping circles are orthogonal to O (not shown).

4.4 Formulæ

The next order of business is to derive useful formulæ for analyzing d(t). Let C(z, r) denote the circle in the complex plane centered at z and of radius $r > 0$. The inversive distance between the circle C(z, r) and its rotated cousin ${\mathfrak {r}}(C(z, r) ) = C({\mathfrak {r}}(z), r)$ is

$$\begin{aligned} h(z,r) = \frac{| z - {\mathfrak {r}}(z)|^{2} - 2 r^{2}}{2r^{2}} = \frac{1}{2}\,\frac{ |\sqrt{3}z - \varvec{i}|^2}{r^{2}} -1. \end{aligned}$$

(3.1)

Fig. 10

The centers of C(z, r) lie on the dotted ellipse with equation $|z-f_1|+|z-f_2|=r_1+r_2$

Let $f_{1}$ and $f_{2}$ be the respective centers of the two circles $A_{1}$ and $A_{2}$. A glance at Fig. 10 should convince the reader that if C(z, r) is externally tangent to $A_{1}$ and internally tangent to $A_{2}$, then $|z - f_{1} | = r_{1} + r$ and $|z - f_{2}| = r_{2} - r$, where $r_{1}$ and $r_{2}$ are the respective radii of the circles $A_{1}$ and $A_{2}$. It follows that

$$\begin{aligned} |z - f_{1} | + |z - f_{2}| = r_{1} + r_{2}, \end{aligned}$$

(3.2)

implying that the centers $z = z(t)$ of the circles $\mu _{t}(C)$ lie on the ellipse described by (3.2). Using $|z - f_{1} | = r_{1} + r$ to eliminate r from (3.1) gives the formula

$$\begin{aligned} h(z) = \frac{1}{2}\,\frac{ |\sqrt{3}z - \varvec{i}|^2}{\big (|z - f_{1} | -r_{1}\big )^{2}} -1 \end{aligned}$$

(3.3)

for the inversive distance between a circle C(z, r) externally tangent to $A_{1}$ and its rotated cousin ${\mathfrak {r}}(C(z,r))$. To find the extrema of the inversive distance function d(t) of (2.2), we need to find the extrema of h(z) of (3.3) when z is subject to the constraint of (3.2).

The ellipse of (3.2) has foci $f_{1}$ and $f_{2}$ with center $(f_{1} + f_{2})/2$, major radius $(r_{1} + r_{2})/2$, and minor radius $\frac{1}{2}\sqrt{(r_{1} + r_{2})^{2} - (f_{2} - f_{1})^{2}}$. The Cartesian equation for this ellipse is

$$\begin{aligned} \frac{(2x - f_{1} - f_{2})^{2}}{(r_{1} + r_{2} )^{2}} + \frac{4y^{2}}{(r_{1} + r_{2})^{2} - (f_{2} - f_{1})^{2}} = 1. \end{aligned}$$

(3.4)

Equation (3.3) has been used to compute the value of $d(\tau )$, the critical minimum value of d(t) in the examples presented in this paper, and the values of $d(t) = d(t')$ for the Ma–Schlenker pairs. We now describe the derivations of the plots of Figs. 7, 8 and 9. In each case this is accomplished by mapping via a Möbius transformation to a standard flow, applying the standard flow, and mapping back.

Fig. 11

The canonical Möbius flow for the parabolic case

4.4.1 The Parabolic Case, Fig. 7

Let $a_{1}= a_{1}(t)$ and $a_{2}=a_{2}(t)$ be the points of tangency of the circle $C(t) = \mu _{t}(C)$ with the respective enveloping circles $A_{1}$ and $A_{2}$, as in Fig. 11. We will derive formulæ for $a_{1}$ and $a_{2}$ in terms of parameters $1< x_{1} < x_{2}$ and the variable t and use these to write the center z(t) and radius r(t) of C(t) for use in the function

$$\begin{aligned} d(t) = h (z(t),r(t)) \end{aligned}$$

(3.5)

of formula (3.1). The Möbius transformation $T(z) = 1/z$, which is its own inverse, maps $A_{1}$ to the vertical line $x= 1/x_{1}$ and $A_{2}$ to the vertical line $x=1/x_{2}$. The flow $\mu _{t}$ is obtained by conjugation of the standard unit speed flow $\nu _{t}(z) = z- t\varvec{i}$ with T. The formulæ for $a_{1}(t)$ and $a_{2}(t)$ become

$$\begin{aligned} a_{s}=a_{s}(t) = T\Bigg (\frac{1- x_{s} t \varvec{i}}{x_{s}}\Bigg ) = x_{s}\, \frac{1+ x_{s} t \varvec{i}}{1 + x_{s}^{2}t^{2}} \quad \text {for }s=1,2. \end{aligned}$$

The radius r(t) of C(t) becomes

$$\begin{aligned} r = r(t) = \frac{r_{1}r_{2}(a_{2}(t) - a_{1}(t))}{r_{2}(a_{1}(t)-f_{1})+ r_{1}(a_{2}(t) - f_{2})}, \end{aligned}$$

(3.6)

and its center z(t) becomes

$$\begin{aligned} z= z(t) = f_{1} + (r_{1} + r(t) )\, \frac{a_{1}(t) - f_{1}}{r_{1}}. \end{aligned}$$

(3.7)

Figure 7 was generated using formulæ (3.6) and (3.7) in (3.5) in Mathematica. We have posted Mathematica notebooks for generating visualizations of these constructions at https://w3.cs.jmu.edu/bowersjc/page/circles/.

Fig. 12

The canonical Möbius flow for the hyperbolic case

4.4.2 The Hyperbolic Case, Fig. 8

In this case the circles $C_{u}$ and $C_{v}$ meet at points $\pm y\varvec{i}$ as in Fig. 12. The relationships among the parameters are that $y < 1/\sqrt{3}$, and for $s=1,2$, $f_{s}^{2} + y^{2} = r_{s}^{2}$ and $x_{s} = f_{s} + r_{s}$. The graphic is computed from (3.5), (3.6), and (3.7) with the only difference from the parabolic case that the formulæ for $a_{1}$ and $a_{2}$ change. To obtain the correct formulæ, we use the Möbius transformation

$$\begin{aligned} T(z) = \frac{x_{2} - y\varvec{i}}{x_{2} + y\varvec{i}}\, \frac{z+ y\varvec{i}}{z- y \varvec{i}} \end{aligned}$$

(3.8)

to map $-y\varvec{i}$ to 0, $y\varvec{i}$ to $\infty$, and $x_{2}$ to 1. The image of $A_{2}$ is the real axis and the image of $A_{1}$ is the line through the origin at angle

$$\begin{aligned} \theta = \cos ^{-1} \Bigg (\frac{f_{1}f_{2} + y^{2}}{r_{1}r_{2}}\Bigg ) \end{aligned}$$

(3.9)

up from the real axis. The standard unit speed flow is now $\nu _{t}(z) = e^{t} z$ and $a_{1}$ and $a_{2}$ are given as

$$\begin{aligned} a_{1} = a_{1}(t) = T^{-1}\big (e^{t+ \theta \varvec{i}}\big ) \quad \text {and}\quad a_{2} = a_{2}(t) = T^{-1} (e^{t}), \end{aligned}$$

(3.10)

where, setting $\kappa = (x_{2} - y\varvec{i})/(x_{2} + y \varvec{i})$, we may write T and $T^{-1}$ as

$$\begin{aligned} T(z) = \kappa \,\frac{z + y \varvec{i}}{z- y \varvec{i}} \quad \text {and}\quad T^{-1} (z) = y\varvec{i}\,\frac{z+ \kappa }{z-\kappa }. \end{aligned}$$

(3.11)

4.4.3 The Elliptic Case, Fig. 9

In this case the circles $C_{u}$ and $C_{v}$ are part of a coaxial family with foci $\pm y$. The relationships among the various parameters are $0< y < 1$, and for $s=1,2$, $f_{s} = (y^{2} + x_{s}^{2})/2x_{s}$ and $r_{s} = x_{s} - f_{s} = (y^{2}-x_{s}^{2})/2x_{s}$. Again, the graphic is computed from (3.5), (3.6), and (3.7) with $a_{1}$ and $a_{2}$ given by

$$\begin{aligned} a_{1} = a_{1}(t) = T^{-1} \big (e^{t\varvec{i} }\big ) \quad \text {and} \quad a_{2} = a_{2}(t) = T^{-1} \big (e^{t\varvec{i}} T(x_{2})\big ). \end{aligned}$$

(3.12)

Here T is the Möbius transformation that takes y to 0, $-y$ to $\infty$, and $x_{1}$ to 1. This takes the circle $A_{1}$ to the unit circle and $A_{2}$ to a circle centered at the origin of radius greater than 1. The standard unit speed flow is the rotation flow $\nu _{t}(z) = e^{t\varvec{i}} z$ and T and $T^{-1}$ are given by

$$\begin{aligned} T(z) = \kappa \,\frac{ z - y }{z + y} \quad \text {and}\quad T^{-1} (z) = -y\,\frac{z+ \kappa }{z-\kappa }, \end{aligned}$$

(3.13)

where $\kappa = (x_{1} + y)/(x_{1} - y)$.

   

Fig. 13

The angle $\alpha$

4.5 The Angular Equation

One more geometric fact is useful. In the setting of the planar construction of Sect. 2.1, let $\alpha = \alpha (t)$ be the angle subtended by the ray from the incenter of the triangle $\Delta$ through the center of $C_{w'} = C_{w'}(t)$ and a ray from the incenter that is tangent to $C_{w'}$, as in Fig. 13. It is easy to see that the angle $\alpha$ lies between 0 and $\pi /2$, and an algebraic manipulation determines that

$$\begin{aligned} d(t) = \langle C_{w'}, {\mathfrak {r}}(C_{w'}) \rangle = \frac{1}{2} + \frac{3}{2} \cot ^{2} \alpha (t). \end{aligned}$$

(3.14)

Notice as $\alpha$ strictly increases from 0 to $\pi /2$, the function $\cot ^{2} \alpha$ strictly decreases from $\infty$ to 0. Thus the comparison of two values of d may be made simply by comparing the corresponding angles $\alpha$:

$$\begin{aligned} d(t) < d(t') \quad \text { if and only if } \quad \alpha (t) > \alpha (t'). \end{aligned}$$

(3.15)

From this observation one can make quick judgements of the correctness of many of our claims in the examples just by examining the graphics of the planar circle realizations of the figure, perhaps with a straight edge and protractor.

Fig. 14

Two views of a state of the construction of a Ma–Schlenker realization with $C_{u}$, $C_{v}$, and $C_{w}$ centered on the equator. (a) Above the north pole. (b) A 3D view

5 The Construction on the Sphere

In this final section we present an alternative description of the construction of Ma–Schlenker realizations, this time directly on the 2-sphere ${\mathbb {S}}^{2}$ realized as the unit sphere in ${\mathbb {E}}^{3} = {\mathbb {C}} \times {\mathbb {R}}$. In this normalization, the north pole is , the south pole is , and the equatorial plane is identified with ${\mathbb {C}}$. The circles $C_{u}$, $C_{v}$, and $C_{w}$ in ${\mathbb {S}}^{2}$ are of equal radii whose centers are equally spaced on the equator. Projecting orthogonally along the north-south direction to the equatorial plane, the upper hemisphere projects to the unit disk and the three circles $C_{u}$, $C_{v}$, and $C_{w}$ project to symmetrically placed chords of the unit circle, as in Fig. 14(a). The orthogonal projections of the typical circles $A_{1}$ and $A_{2}$ in the coaxial family ${\mathcal {A}}_{C_{u}, C_{v}}$ that form the envelope of the Möbius-flowed circles $C_{w'}(t)$ are shown in Fig. 14(a). The coaxial family ${\mathcal {A}}_{C_{u},C_{v}}$ is obtained as follows. Let $\Pi _{u}$ and $\Pi _{v}$ be the 2-planes in ${\mathbb {E}}^{3}$ whose respective intersections with the 2-sphere ${\mathbb {S}}^{2}$ are the circles $C_{u}$ and $C_{v}$. Let $\ell _{u,v}$ be the line $\Pi _{u} \cap \Pi _{v}$, a line that runs in the north-south direction. Then ${\mathcal {A}}_{C_{u},C_{v}}$ is precisely the collection of circles formed as intersections $\Pi \cap {\mathbb {S}}^{2}$ as $\Pi$ ranges over all 2-planes in ${\mathbb {E}}^{3}$ that contain the line $\ell _{u,v}$. Notice that the circles in the coaxial family ${\mathcal {A}}_{C_{u},C_{v}}$ project orthogonally to the family of lines in ${\mathbb {C}}$ that pass through the common point q, the intersection of the vertical line $\ell _{u,v}$ with the equatorial plane ${\mathbb {C}}$. This explains the position of the projections of $A_{1}$ and $A_{2}$ in Fig. 14(a). In this set up, local maxima for d(t) occur when the three circles corresponding to $u'$, $v'$ and $w'$ are all centered on the equator. This occurs once when the flow $\mu _{t}$ is hyperbolic or parabolic, and periodically with two different local maximum values as the flow pushes the circle across the equator when elliptic. By shifting the origin, we may assume that a local maximum occurs at time $t=0$ so that the circles $C_{u'}(0)$, $C_{v'}(0)$, and $C_{w'}(0)$ are centered on the equator, and, without loss of generality, we may assume that the flow is oriented so that $C_{w'}(t)$ is centered in the upper hemisphere for initial positive values of t. Under these normalizations $d(t) = d(-t)$ and $C_{w'}(-t) = I_{O}(C_{w'}(t))$ for all t, where O is the equator. The resulting realizations of the Ma–Schlenker octahedra of the form ${\mathscr {O}}(a,b,c,d)$ have order three rotational symmetry about the axis through the north and south poles. We should comment that a realization built in this manner will not be a circle packing. To obtain a packing these will need to be Möbius flowed using a hyperbolic flow from the north toward the south pole.

Fig. 15

The view of a critical circle realization for ${\mathscr {O}}(1,b,1,1)$ from above the north pole, with $C_{u}$, $C_{v}$, and $C_{w}$ centered on the equator

As an example, we construct a critical circle packing for the Ma–Schlenker octahedron ${\mathscr {O}}(1, b,1,1)$ where $b>1$ and $d(\tau ) = 1$. Choose $C_{u}$, $C_{v}$, and $C_{w}$ to have equal radii of $\pi /3$ so that the three circles are mutually tangent and project orthogonally to an equilateral triangle inscribed in the unit circle, as in Fig. 15. Choose $A_{1} = C_{v}$ and $A_{2}$ so that the circles $C_{u'}(t)$, $C_{v'}(t)$, and $C_{w'}(t)$ are disjoint except at two values $t = \pm \tau$ of the flow variable, when the circles are mutually tangent. Automatically, d(t) must take on its isolated minimum values at $t=\pm \tau$ where $d(\pm \tau ) =1$, with $d(t) >1$ for $t \ne \pm \tau$. To obtain a critical Ma–Schlenker c-octahedron, apply the hyperbolic Möbius flow from the north toward the south pole whose flow lines are the meridianal circles until the centers of $C_{u}$, $C_{v}$, and $C_{w}$ lie on a latitude L and those of $C_{u'}(\tau )$, $C_{v'}(\tau )$, and $C_{w'}(\tau )$ lie on a latitude $L'$ of equal radii, L centered on the south pole and $L'$ on the north. The resulting realization is a circle packing and therefore a critical Ma–Schlenker c-octahedron.