Orthogonal ring patterns in the plane

We introduce orthogonal ring patterns consisting of pairs of concentric circles generalizing circle patterns. We show that orthogonal ring patterns are governed by the same equation as circle patterns. For every ring pattern there exists a one parameter family of patterns that interpolates between a circle pattern and its dual. We construct ring patterns analogues of the Doyle spiral, Erf and zα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z^\alpha $$\end{document} functions. We also derive a variational principle and compute ring patterns based on Dirichlet and Neumann boundary conditions.


Introduction
The theory of circle patterns can be seen as a discrete version of conformal maps.Schramm [6] has studied orthogonal circle patterns on the Z 2 -lattice, has proven their convergence to conformal maps and constructed discrete analogs of some entire holomorphic functions.Circle patterns are described by a variational principle [5], which is given in terms of volumes of ideal hyperbolic polyhedra [4].We introduce orthogonal ring patterns that are natural generalizations of circle patterns.Our theory of orthogonal ring patterns has its origin in discrete differential geometry of S-isothermic cmc surfaces [3].Recently, orthogonal double circle patterns (ring patterns) on the sphere have been used to construct discrete surfaces S-cmc by Tellier et al. [7].
We start Sect. 2 with a definition of orthogonal ring patterns and their elementary properties.In particular we show that all rings have the same area.Our main Theorem 2.4 shows that ring patterns are described by an equation for variables at the vertices.Furthermore, each ring pattern comes with a natural 1-parameter family of patterns.In Sect. 3 we show that as the area of the rings goes to zero the ring patterns converge to orthogonal circle patterns.In the following Sect.4 we introduce ring patterns analogs of Doyle spirals, the Erf function, z α for α ∈ (0, 2], and the logarithm.Finally, we introduce a variational principle to construct ring patterns for given Dirichlet or Neumann boundary conditions.A remarkable fact that we explore is that the orthogonal ring and circle patterns in R 2 are governed by the same integrable equation.In a subsequent publication we plan to develop a theory of ring patterns in a sphere and hyperbolic space.They are governed by equations in elliptic functions that belong to the class of discrete integrable systems classified in [1].

Orthogonal ring patterns
In this section, we will introduce orthogonal ring patterns and show that the existence of such the patterns is governed by the same equation as the existence of orthogonal circle patterns.
We will consider cell complex G defined by a subset of the quadrilaterals of the Z 2 lattice in R 2 .The vertices V (G) of the complex G are indexed by (m, n) ∈ Z 2 and denoted by v m,n .Each of its inner vertices has four neighbors, the vertices with less neighbors are called boundary vertices.The vertices of the dual cell complex G * are identified with the 2-cells of G, and the edges of G * correspond to the inner edges of G, i.e. to the edges shared by two neighboring 2-cells.We assume that G and G * are simply connected.The oriented edges are given by pairs of vertices and are either horizontal A ring is a pair of two concentric circles in R 2 that form a ring (annulus).We identify the vertices with the centers and denote the inner circle and its radius by small letters c and r, and the outer circle and its radius by capital letters C and R. We assign an orientation to the ring by allowing r to be negative: positive radius corresponds to counter-clockwise and negative radius to clockwise orientation.The outer radius will always be positive.The area of a ring is given by (R 2 − r 2 )π.Subscripts are used to associate circles and radii to vertices of the complex, e.g., c m,n is the inner circle associated with the vertex v m,n .The orthogonal intersection of neighboring rings has the following implication for their areas.
Lemma 2.2.Consider two rings with radii r i , R i and r j , R j that intersect orthogonally.Then the two rings have the same area.
Proof.By Pythagoras' Theorem the square of the distance d between the circle centers is R 2 i + r 2 j = d 2 = r 2 i + R 2 j since the inner and outer circles are intersecting orthogonally.This equation is equivalent to the equality of the ring areas (R  The constant area allows us to use a single variable ρ i to express the inner and the outer radii of the rings in the following way: Consider an orthogonal ring pattern with constant ring area A 0 = πℓ 2 0 , that is, for the radii Then for each vertex we can choose a single variable ρ i by setting (1) R i = ℓ 0 cosh(ρ i ) and r i = ℓ 0 sinh(ρ i ) .
We will call those new variables ρ-radii.The orientation of the rings is encoded in the sign of the ρ-radii.In Sect. 3 we consider the limit of orthogonal ring patterns as the area goes to zero.The ρ-radii become the logarithmic radii of a Schramm type orthogonal circle pattern [6] in the limit.
As in the case of orthogonal circle patterns there exist families of vertices such that all rings along the diagonals touch (see Fig. 2).
Neighboring vertices of an orthogonal ring pattern define a cyclic quadrilaterals of the following forms: The circles C i , c i and C j , c j intersect in four points.Since the inner circle c i (resp.c j ) and the outer circle C j (resp.C i ) intersect orthogonally the centers of the circles and the intersection points c i ∩ C j and C i ∩ c j lie on a circle.We introduce four possible circular quadrilaterals, shown in Fig. 3, depending on the orientation of the rings (i.e. on the sings of the ρ-radii).Note that, the angle at the vertex v i has the same sign as the corresponding ρ i .
If ρ i = 0 the inner circle c i shrinks to its center and the cyclic quadrilateral defined by the rings (C i , c i ) and (C j , c j ) degenerates to a triangle with a double vertex.The circle C j passes through this point.
Given the ρ-radii we can compute the angles in the cyclic quadrilaterals.We will assume that the arctan function maps to oriented angles in (− π 2 , π 2 ).Lemma 2.3.Let v i and v j be two neighboring vertices in an orthogonal ring pattern with ρ-radii ρ i and ρ j .Then the angle at the vertex v i in the quadrilateral (triangle if ρ i = 0) defined by the two rings at v i and v j is given by Proof.For ρ i ̸ = 0 the angle φ ij is built by two angles of two rectangular triangles Simple transformations of hyperbolic functions yield Further, using we arrive at the representations (2) for all ρ j .The angle φ ij is discontinuous at ρ i = 0, and its value jumps by π: For ρ i = 0 the circle c i degenerates to a point located at the center of C i , and the circle C j passes through this point.The quadrilateral degenerates to a triangle, and the angle of this triangle at the vertex v i is

□
We define a cone angle at v i as the sum of the angles built by the ring centered at v i with all its neighbors: For interior vertices of an orthogonal ring pattern we have For a boundary vertex Θ i > 0 if it is positively oriented ρ i > 0, and Θ i < 0 if it is negatively oriented ρ i < 0.
Theorem 2.4 (Orthogonal ring patterns).An orthogonal ring pattern R with simply connected G and G * is uniquely determined by its ρ-radii function ρ : V (G) → R.
A function ρ : V (G) → R describes the ρ-radii of an orthogonal ring pattern on G with the boundary cone angles Θ i if and only if it satisfies: (4) Here the sum is taken over all neighboring vertices of v i , and Val(i) is the number of rings neighboring to the boundary ring i.
Proof.The first claim of the theorem follows from the fact that a pair of orthogonal rings is determined by their ρ-radii uniquely up to Euclidean motion.Consequently laying the rings we obtain a simply connected ring pattern.
Let v i ∈ V (G) be an interior vertex with four neighboring vertices v 1 , v 2 , v 3 , and v 4 .The five rings form a flower in the pattern if and only if the angles φ ij for j ∈ {1, 2, 3, 4} sum up to 2π (or −2π, depending on the orientation).By Lemma 2.3 for positive ρ i the sum of the angles This is equivalent to (4).For negative ρ i the other equation of Lemma 2.3 also implies (4).Hence we can assemble the four quadrilaterals and rings around the vertex v i to form an orthogonal ring pattern.As the complex G is simply connected the local proof suffices to prove that the entire complex G can be assembled to build an orthogonal ring pattern.
The ρ-radii satisfy the same equation (4) for the cases ρ i > 0 and ρ i < 0. This equation is also satisfied for ρ i = 0.This can be seen as the limit ρ i → 0 since the right hand side of ( 4) is a continuous function of ρ i .Alternatively, when the quadrilaterals degenerate to triangles the angles of the triangles at the vertex v i are given by (2) in the case ρ i = 0. Summing up around v i and using Θ i = 0 we arrive at the same equation ( 4).
Formulas for the cone angles at the boundary rings follow directly from (2). □ The angle condition at the vertices of Thm.2.4 only depends on the differences of the logarithmic radii.So without violating equation ( 4), we can apply a shift ρ → ρ δ = ρ + δ by δ ∈ R to the ρ-variables.
Corollary 2.5.Consider an orthogonal ring pattern R of area π for given ρ-radii ρ i .Then the ρ-radii ρ δ i = ρ i + δ define a one parameter family of orthogonal ring patterns R δ with radii: and area A δ = π.

Relation to orthogonal circle patterns
In this section we give a detailed description of the relation of orthogonal ring patterns and orthogonal circle patterns.It turns out that orthogonal circle patterns can be considered as a special case of ring patterns with constant ring area A 0 = 0.
To formulate the limit we need to review some properties of orthogonal circle patterns.Two orthogonally intersecting circles in an orthogonal circle pattern create a cyclic right angled kite (see Fig. 5 left and right).The angle φ • ij at a vertex v i in a kite on the edge (v i , v j ) of an orthogonal circle pattern with radii r • i = e ρ i is given by: (5) In case of circle patterns the ρ-radii are called logarithmic radii.Logarithmic radii of an immersed orthogonal circle pattern are governed by the same equation (cf.[6,5]) as the ρ-radii of ring patterns (see Thm. 2.4).Furthermore, for each orthogonal circle pattern C with logarithmic radii ρ i there exists a dual pattern C * with radii e −ρ i .The angles of the dual pattern are given by Note that the angles at interior vertices still sum up to 2π, but the angles at the boundary vertices change as shown in Fig. 4. Now let us go back to the one parameter family R δ of ring patterns defined in Cor.2.5.To avoid that the radii go to infinity as δ → ±∞ we scale the entire pattern by 2e −|δ| .So the radii of the one parameter family of ring patterns are: r δ m,n = 2e −|δ| sinh(ρ m,n + δ) and R δ m,n = 2e −|δ| cosh(ρ m,n + δ).In the limit δ → ±∞ the areas of the rings tend to zero and for the radii we have: Remark 3.0.(Limits on compact subsets).If the ring pattern R is infinite we consider the limits δ → ±∞ of the family R δ on any compact subset G 0 ⊂ G satisfying the same conditions as G, i.e.G 0 and G * 0 are simply connected.
Limit δ → +∞.For δ > − min v i ∈G 0 ρ i we have ρ δ i = ρ i + δ > 0 for all v i ∈ G 0 .So considering the limit as δ → ∞ all ρ δ i will be positive and the angles of the circle pattern C (equation ( 5)) are exactly those of the ring pattern R δ given in Lemma 2.3.Furthermore, for δ → ∞, we obtain rings with area 0 since the outer and inner radii both converge to e ρ i .The neighboring circles intersect orthogonally because inner and outer circles of the orthogonal ring pattern are intersecting orthogonally in the entire one parameter family.The limit circles form a Schramm type orthogonal circle pattern.
Limit δ → −∞.For δ < − max v i ∈G 0 ρ i all ρ δ i = ρ i +δ < 0. By Lemma 2.3 the angles of the ring pattern for negative ρ i are given by and correspond to the angles of the dual pattern C * with opposite orientation.As equation ( 4) is satisfied for all δ, we obtain the dual orthogonal circle pattern C * (with opposite orientation) in the limit.
Corollary 3.1.Let R δ be a one parameter family of orthogonal ring patterns with ρ δ i = ρ i +δ for ρ i ∈ R as described in Cor.2.5.Then for δ → +∞ we obtain an orthogonal circle pattern C with logarithmic radii ρ i and for δ → −∞ we obtain the dual circle pattern C * with logarithmic radii −ρ i .
Here the limits are understood in the sense of Remark 3.0.For a better understanding of the deformation, the one parameter family of cyclic quadrilaterals associated to a single edge (v i , v j ) is shown in Fig. 5: Assume that ρ i and ρ j are both positive and ρ i < ρ j .Then the deformation starts with an embedded cyclic quadrilateral (center right).For δ → ∞ we obtain two orthogonally intersecting circles with radii e ρ i and e ρ j that form a kite (bottom right).When δ ↘ −ρ i one of the edges at v i shrinks to a point and reverses its direction as ρ i + δ changes its sign from + to −.If −ρ j < δ < −ρ i then r δ i < 0 and we obtain a non-embedded quadrilateral (top center).Again as δ ↘ −ρ j one edge at v j shrinks to a point and changes its direction as ρ j + δ changes sign (center left) and we obtain an embedded quadrilateral with negative orientation.For δ → −∞ the areas of the rings go to zero and we obtain two orthogonally intersecting circles with radii e −ρ i and e −ρ j (bottom left).

Doyle spiral, Erf, and z α ring patterns
In this section we will have a look at some known orthogonal circle patterns and consider their ring pattern analogs and deformations.4.1.Doyle spirals.Doyle spirals for the square lattice have been constructed by Schramm [6].For x + iy ∈ C \ {0} Schramm defines radii by r m,n = |e (x+iy)(m+in) |.Taking the logarithm we obtain the logarithmic radii ρ m,n = mx − ny.We will take these radii as a definition of the Doyle spiral ring pattern.
Let us consider the generic case x y ̸ ∈ Q when the ρ-radii do not vanish.By Lemma 2.3 the angles of the cyclic quadrilaterals at the edges are given by Consider the one parameter family R δ of orthogonal ring patterns as described by Cor.2.5.The angles along the horizontal and vertical edges stay constant in the two halfspaces.As in the general case discussed in the previous section, all ρ's become positive for δ → +∞ (resp.negative for δ → −∞), see Remark 3.0, and we obtain a Doyle spiral and its dual as constructed by Schramm (see Fig. 6).4.2.Erf pattern.For analogs to Schramm's √ i-Erf pattern let us have a look at the corresponding radius function given in [6] r m,n = e amn for (m, n) ∈ Z 2 and a ∈ R, a > 0. Taking the logarithm we obtain ρ m,n =  The angles in the pattern are given by As ρ m,n = amn the ρ-radii change signs at the coordinate axes.In the four quadrants, the angles along the horizontal and the vertical parameter lines are constant.All the rings on the coordinate axes are congruent: the radii of their outer circles are equal to R = cosh 0 = 1, and their inner circles degenerate to their centers.
If we consider the one parameter family of ring patterns defined in Cor.2.5 we see that in the limit δ → +∞ we obtain the √ i-SG Erf circle patterns constructed by Schramm.For δ → −∞ we obtain a pattern with ρ * m,n = −amn.This is the same pattern as for a since ρ * m,n = ρ −m,n .
4.3.z α and logarithm patterns.In [2] the authors defined an orthogonal circle pattern C(z α ) as a discretization of the complex map z → z α for α ∈ (0, 2).The radius function of the circle pattern is given by the following identities (cf.[2, Thm. 3, equation (10, 11)]) on a subset of Z 2 given by It is known that the dual pattern of z α is given by z 2−α , e.g., the dual circle pattern of C(z 2/3 ) is C(z 4/3 ) = (C(z 2/3 )) * shown in Fig. 8 (top left and bottom right).Based on the logarithmic radii of these patterns we construct a one parameter family of ring patterns that interpolates between the two patterns.
An orthogonal circle pattern for z 2 can be defined by considering a special limit for α → 2. The radii of the z 2 pattern are defined in [2,Sect. 5].The dual of z 2 is the logarithm map log z.In each of the corresponding orthogonal circle patterns, one of the circles degenerates.In case of z 2 one of the circles has radius 0, i.e., the circle degenerates to a point and the logarithmic radius is negative infinity.Consequently, one of the circles in the log z pattern has radius infinity, i.e., the circle degenerates to a line and the logarithmic radius is positive infinity.We illustrate the one parameter deformation of z 2 to log(z) in Fig. 9.

Variational description
The construction of a ring pattern is very similar to the construction of an orthogonal circle pattern since the equations at the interior vertices are the same (see Thm. 2.4).For (not necessarily orthogonal) circle patterns there exists a convex variational principle [5].For planar orthogonal circle condition (7) implies that the solutions possess the symmetry group (6) described in Section 3.

□
An example of solution of a Neumann boundary value problem is presented in Fig. 10).Here for all interior vertices Φ = 2π, for all boundary and not corner vertices Φ i = Θ i = π.Four angles Φ i = Θ i at corner vertices of the quadrilateral should sum up to 2π.One can easily check that the last condition implies (7).

Definition 2 . 1 (( 1 )
Orthogonal ring patterns).An orthogonal ring pattern consists of rings associated to the vertices of G satisfying the following properties: The rings associated to neighboring vertices v i and v j intersect orthogonally, i.e., the outer circle C i of the one vertex intersects the inner circle c j of the other vertex orthogonally and vice versa (see Fig. 1, left).(2) In each square of G the inner circles c m,n and c m+1,n+1 and the outer circles C m,n+1 and C m+1,n pass through one point.(Then orthogonality implies that the two inner and the two outer circles touch in this point.see Fig. 1, center).(3) For any ring (C m,n , c m,n ) the four touching points C m,n ∩ C m+1,n−1 , c m,n ∩ c m+1,n+1 , C m,n ∩ C m−1,n+1 and c m,n ∩ c m−1,n−1 have the same orientation as c m,n , i.e., are in counter-clockwise order if r m,n is positive and in clockwise order if r m,n is negative.

Figure 1 .
Figure 1.Left: Two orthogonally intersecting rings.Center: The inner circles touch along one diagonal of a quadrilateral and the outer circles along the other diagonal.The touching point coincides.Right: If the orientation (i.e., signed radii) of the inner circles differ, then the centers lie on the same side of the common tangent.

Figure 2 .
Figure 2. The rings of an orthogonal ring pattern partition into two diagonal families of touching rings.

Figure 4 .
Figure 4.An orthogonal circle pattern and its dual.The boundary angles the dual pattern are 2π − φ ij resp.π − φ ij depending on whether the degree of the boundary vertex is 3 or 2.

Proposition 4 . 1 (
Doyle spiral ring pattern).Let x + iy ∈ C \ {0} be a complex number.The Doyle spiral ring pattern is given by the

Figure 5 .
Figure 5. Deformation of a cyclic quadrilateral defined by two orthogonally intersecting rings.The bottom left and bottom right show the limits of the ring pattern as the area of the ring goes to zero.Positive radii are indicated by orange, negative radii (i.e., negative ρ) are indicated by pink circles.The angle associated with the left vertex is shown in green.

Figure 6 .
Figure 6.Deformation of an orthogonal circle pattern (top left) into its dual (bottom right) through a one parameter family of ring patterns (top right and bottom left).We see how the orientation of the quadrilaterals flips during the deformation.The innermost vertex in the top left circle patterns becomes the outermost vertex in the bottom right circle pattern.

Figure 7 .
Figure 7.An Erf ring pattern (left) and the corresponding limit circle pattern.

Figure 8 .
Figure 8.One parameter family of orthogonal ring patterns interpolating between the orthogonal circle pattern for z → z 2/3 (top left) and the dual pattern for z 4/3 (bottom right)

Figure 9 .
Figure 9. Orthogonal ring patterns interpolating between the circle patterns for z 2 and its dual pattern for log z.

Figure 10 .
Figure10.An orthogonal ring pattern computed using the variational principle with Neumann boundary conditions.The prescribed angles are π for the boundary vertices of degree 2. The shape is governed by the four angles, with the sum 2π, prescribed for the four corner boundary vertices of degree 1.