Abstract
While a generic smooth Ribaucour sphere congruence admits exactly two envelopes, a discrete Rcongruence gives rise to a 2parameter family of discrete enveloping surfaces. The main purpose of this paper is to gain geometric insights into this ambiguity. In particular, discrete Rcongruences that are enveloped by discrete channel surfaces and discrete Legendre maps with one family of spherical curvature lines are discussed.
Introduction
Classically, a smooth 2parameter family of spheres is called Ribaucour sphere congruence if its two enveloping surfaces have corresponding curvature lines. Since a smooth sphere congruence admits at most two envelopes, each such sphere congruence provides a Ribaucour pair of surfaces. Over the decades, this transformation concept was extended to submanifolds, partly also with singularities, [15,16,17, 28, 30] and various (integrable) approaches for the construction were discussed [8, 18, 31]. In these developments, special interest was paid to constrained Ribaucour transformations that allow to generate envelopes with special geometric properties, for example, the Darboux transformation of isothermic surfaces and Ribaucour transformations that preserve classes of Osurfaces [26].
A characteristic feature in the theory of smooth Ribaucour transformations are permutability theorems. In discrete differential geometry, those became crucial as underlying concept of integrable discretizations of curvatureline parametrized surfaces and orthogonal coordinate systems [4, 6].
Discrete equivalents of Ribaucour sphere congruences were developed in [6] and [19]: discrete Rcongruences are provided by Qnets in the Lie quadric, that is, a discrete congruence of spheres with planar faces. However, more flexibility in the discrete setup results in a 2parameter family of discrete envelopes; we call it the Ribaucour family of a discrete Rcongruence (cf. Definition 3.7).
In the last years, in a variety of works, discrete Ribaucour transformations between general discrete surfaces were discussed [9, 20, 25]. Furthermore, restricted transformations for particular classes of discrete surfaces as discrete (special) isothermic surfaces [4, 11, 12, 22, 23] and discrete Osurfaces [29] were investigated, to name just a few.
However, to the best of the authors’ knowledge, there are no systematic studies of the geometry of the entire Ribaucour families in the literature, since usually pairs of discrete envelopes are considered as the main objects. Geometric investigations of this 2parameter family of envelopes is the main contribution of this work. In most parts we consider the discrete Rcongruences as the primary objects of interest and explore the corresponding Ribaucour families by using data provided by the sphere congruence.
This paper is organized as follows. In Sect. 2 we recall basic principles and constructions in Lie sphere geometry. Section 3 is devoted to the construction of envelopes of discrete Rcongruences. As a crucial result, for any face of a discrete Rcongruence, we introduce two involutive Lie inversions that map adjacent contact elements of discrete envelopes onto each other (Proposition 3.2). Using these inversions, we obtain a construction of a unique envelope from one prescribed initial contact element. Hence, in this way, we can parametrize the entire Ribaucour family of a discrete Rcongruence. To conclude this section, we discuss some examples of special discrete Rcongruences. This reveals relations to recent works on discrete Ribaucour coordinates [9] and discrete \(\Omega \)surfaces [7]. Since we consider the discrete Rcongruence as primary object, our approach sheds light on these situations from a different point of view.
In Sect. 4 we focus on the geometry of the envelopes in a general Ribaucour family. We discuss how the choice of a facewise constant Lie inversion decomposes a Ribaucour family into pairs of envelopes (cf. Sect. 4.2). Those Lie inversions then interact well with facecyclides of these Ribaucour pairs and, therefore, induce a Ribaucour transformation of cyclidic nets as pointed out in Sect.4.3.
As an application of the developed framework for discrete Rcongruences, we investigate envelopes with one family of spherical curvature lines in Sect. 5. In particular, we characterize sphere congruences of Ribaucour families containing at least two discrete channel surfaces.
Preliminaries
In this section we briefly summarize some basic principles of Lie sphere geometry and recall some concepts that will be crucial for the rest of this work. For more details on this topic, the interested reader is referred to Blaschke [1] and Cecil [13].
Throughout the paper we use the hexaspherical model introduced by Lie and work in the vector space \({\mathbb {R}}^{4,2}\) endowed with the inner product \(\langle {\cdot , \cdot }\rangle \) given by
The projective light cone will be denoted by \({\mathbb {P}}({\mathcal {L}})\) and represents the set of oriented 2spheres in \({\mathbb {S}}^3\) in this model. Two spheres \(r,s \in {\mathbb {P}}({\mathcal {L}})\) are in oriented contact if and only if \(\langle {{\mathfrak {r}}, {\mathfrak {s}}}\rangle =0\), where \({\mathfrak {r}}\) and \({\mathfrak {s}}\) denote homogeneous coordinates of the spheres r and s, respectively. Thus, the set \({\mathcal {Z}}\) of lines in \({\mathbb {P}}({\mathcal {L}})\) corresponds to contact elements.
Homogeneous coordinates of elements in the projective space \({\mathbb {P}}({\mathbb {R}}^{4,2})\) will be denoted by the corresponding black letter. If statements hold for arbitrary homogeneous coordinates we do this without explicitly mentioning it.
By breaking symmetry, we can recover various subgeometries of Lie sphere geometry. Thus, let \({\mathfrak {p}} \in {\mathbb {R}}^{4,2}\) be a vector that is not lightlike, i.e., \(\langle {{\mathfrak {p}},{\mathfrak {p}}}\rangle \ne 0\). If \({\mathfrak {p}}\) is timelike, then \(\langle {\mathfrak {p}}\rangle ^\perp \cong {\mathbb {R}}^{4,1}\) defines a Riemannian conformal geometry, that is, a Möbius geometry. In this case, elements in \({\mathbb {P}}({\mathcal {L}})\cap \langle {\mathfrak {p}} \rangle ^\perp \) are considered points and will be called point spheres. If \({\mathfrak {p}}\) is spacelike, it determines a Lorentzian conformal geometry \(\langle {\mathfrak {p}}\rangle ^\perp \cong {\mathbb {R}}^{3,2}\) or planar Lie geometry.
Linear systems and linear sphere complexes
Let W be a nondegenerate, indefinite projective plane, then the 1parameter family \(W \cap {\mathbb {P}}({\mathcal {L}})\) of spheres is called a linear system^{Footnote 1}. If this family of spheres lies in the span of the three spheres \(s_1, s_2, s_3 \in {\mathbb {P}}({\mathcal {L}})\), the Lie invariant
provides information about the geometric configuration:

If \(\delta =1\), then W is a (2, 1)plane. Hence, the spheres of the linear system are curvature spheres of a Dupin cyclide and there exists a 1parameter family of spheres in oriented contact with all spheres of the linear system—the spheres of the orthogonal (2, 1)plane.

If \(\delta =1\), then W is a (1, 2)plane. Thus, there does not exist a sphere that is in oriented contact with all spheres of the linear system, since the orthogonal complement is a (3, 0)plane that does not contain any spheres.
Any element \(a \in {\mathbb {P}}({\mathbb {R}}^{4,2})\) defines a linear sphere complex \({\mathbb {P}}({\mathcal {L}}) \cap a^\perp \), a 3dimensional family of 2spheres. Depending on the type of the vector a, we distinguish three cases:

parabolic linear sphere complex, \(\langle { {\mathfrak {a}}, {\mathfrak {a}}}\rangle =0\): all spheres in the complex are in oriented contact with the sphere defined by a; hence, all spheres in the contact elements that contain the sphere a provide the linear sphere complex.

hyperbolic linear sphere complex, \(\langle { {\mathfrak {a}}, {\mathfrak {a}}}\rangle < 0\): if we fix a Möbius geometry by choosing a as the point sphere complex, then the linear sphere complex consists of all point spheres.

elliptic linear sphere complex, \(\langle { {\mathfrak {a}}, {\mathfrak {a}}}\rangle > 0\): in a Möbius geometry, these linear sphere complexes then consist of all contact elements that intersect a fixed sphere at a fixed angle (see Fig. 1, left).
Since elliptic linear sphere complexes will be essential in this paper, we will discuss their geometric construction illustrated in Fig. 1 in more detail. Assume that \(a \in {\mathbb {P}}({\mathbb {R}}^{4,2})\) determines an elliptic linear sphere complex. Additionally fix a parabolic complex q, as well as a hyperbolic complex p to distinguish a Möbius geometry modelled on \(p^\perp \).
Without loss of generality, we choose \({\mathfrak {q}}=(1,1,0,0,0,0)\), \({\mathfrak {p}}=(0,0,0,0,0,1)\) and homogeneous coordinates \({\mathfrak {a}} = ({\mathfrak {a}}_i)_i \in {\mathbb {R}}^{4,2}\) of a such that \(\langle {{\mathfrak {a}}, {\mathfrak {q}}}\rangle =1\). By defining
we obtain two concentric spheres \(s_r\) and \(s_R\) with center c and radii r and R, respectively (see Fig. 1, left). Then a straightforward computation shows that any sphere s in the elliptic linear sphere complex \(\langle {{\mathfrak {a}}}\rangle ^\perp \) intersects the sphere \(s_R\) under the constant oriented angle \(\cos \gamma = \frac{r}{R}\). If \(\langle {{\mathfrak {a}}, {\mathfrak {q}}}\rangle =0\), the sphere \(s_R\) corresponds to a plane and the sphere \(s_r\) becomes the point at infinity.
Lie inversions
Let \(a \in {\mathbb {P}}({\mathbb {R}}^{4,2})\), \(\langle { {\mathfrak {a}}, {\mathfrak {a}}}\rangle \ne 0\), then the Lie inversion with respect to the linear sphere complex determined by a is given by
Lie inversions are involutive linear maps that preserve oriented contact between spheres and, therefore, also contact elements. Moreover, from the definition, we directly conclude that spheres contained in the linear sphere complex \({\mathbb {P}}({\mathcal {L}}) \cap a^\perp \) are fixed by the Lie inversion \(\sigma _a\).
Lemma 2.1
Let \(r, s \in {\mathbb {P}}({\mathcal {L}})\), \(\langle {{\mathfrak {r}}, {\mathfrak {s}}}\rangle \ne 0\), be two spheres that do not lie in the linear sphere complex \({\mathbb {P}}({\mathcal {L}}) \cap a^\perp \) with \(\langle {{\mathfrak {a}}, {\mathfrak {a}}}\rangle \ne 0\). Then, the \(\text {span} \{ r, s, \sigma _{a}(r), \sigma _{a}(s) \} \cap {\mathbb {P}}({\mathcal {L}})\) is a linear system.
The following two constructions give Lie inversions that map prescribed spheres onto each other:
Lemma 2.2
Let \(\langle {{\mathfrak {r}}}\rangle = r\) and \(\langle {\bar{{\mathfrak {r}}}}\rangle = {\bar{r}}\) be fixed homogeneous coordinates of two spheres that are not in oriented contact and let \(\sigma _\lambda \) denote the Lie inversion with respect to the linear sphere complex \({\mathfrak {n}}_\lambda := {\mathfrak {r}}  \lambda \bar{{\mathfrak {r}}}\), where \(\lambda \in {\mathbb {R}}^\times \). Then the following properties hold:

(i)
For any \(\lambda \in {\mathbb {R}}^\times \), the Lie inversion \(\sigma _\lambda \) maps the sphere r to the sphere \({\bar{r}}\).

(ii)
All spheres in oriented contact with r and \({\bar{r}}\) are fixed by any Lie inversion \(\sigma _\lambda \).

(iii)
A contact element that contains r is mapped by any Lie inversion \(\sigma _\lambda \), \(\lambda \in {\mathbb {R}}\), to the same contact element; however, different Lie inversions induce different correspondences between the spheres in the two contact elements.
Proof
The properties (i) and (ii) follow directly from equation (1). To prove the third statement, we suppose that \(v \in {\mathbb {P}}({\mathcal {L}})\) is a sphere in oriented contact with the sphere r and \(\langle {{\mathfrak {v}}, \mathfrak {{\bar{r}}}}\rangle \ne 0\). Then, the sphere
lies in the contact element \(f:=\text {span} \{ r, v \}\) and, furthermore, for any \(\lambda \in {\mathbb {R}}\) in the linear sphere complex \(\langle {\mathfrak {n}}_\lambda \rangle ^\perp \). Hence, any Lie inversion \(\sigma _\lambda \) preserves the sphere \({\tilde{s}}\) and maps the contact element f to the contact element \(\sigma _\lambda (f)=\text {span} \{ {\tilde{s}}, {\bar{r}} \}\). \(\square \)
Lemma 2.3
Let f and \({\bar{f}}\) be two contact elements sharing a common sphere \(s \in {\mathbb {P}}({\mathcal {L}})\). Then, for four spheres \(r, t \in f\) and \({\bar{r}}, {\bar{t}} \in {\bar{f}}\) that do not coincide with s, there exists a unique Lie inversion \(\sigma \) satisfying
Proof
By assumption, there exist constants \(\lambda , \mu , {\bar{\lambda }}, {\bar{\mu }} \in {\mathbb {R}}^\times \) such that
Then, the Lie inversion with respect to the linear sphere complex \(a \in {\mathbb {P}}({\mathbb {R}}^{4,2})\) determined by
provides the soughtafter map. Moreover, a is the intersection of the lines \(\text {span} \{ r,{\bar{r}} \}\) and \(\text {span} \{ t,{\bar{t}} \}\) and hence unique. \(\square \)
For later reference, we remark the following property for the composition of two Lie inversions that follows from straightforward computations:
Lemma 2.4
Two Lie inversions commute, \(\sigma _{a} \circ \sigma _{b}=\sigma _{b} \circ \sigma _{a}\), if and only if the corresponding linear sphere complexes are involutive, that is, \(\langle {{\mathfrak {a}},{\mathfrak {b}}}\rangle =0\).
Crossratio of four spheres
Similar to the crossratio of four concircular points, one defines the crossratio of four spheres in a linear system. We recall this definition and different ways to compute it.
Definition 2.5
Let \(r_1, r_2, r_3, r_4 \in {\mathbb {P}}({\mathcal {L}})\) be four spheres in a common contact element, then the crossratio is defined by
It is independent of the choice of homogeneous coordinates and can be equivalently described by the four radii of the spheres
where \(R_i\) denotes the radius of the sphere \(r_i\) (for point spheres the radius is assumed to be 0 and for a plane we define the radius as \(\infty \)).
Moreover, one can define the crossratio of four spheres in a linear system by tracking it back to the crossratio of four spheres in a contact element: assume that \(r_1, r_2, r_3\) and \(r_4\) are four spheres in a linear system and choose an arbitrary contact element \(f \ni r_1\). Then there exist three unique spheres \(s_i\) in f such that \(s_i \perp r_i\) for \(i=2,3,4\) and we define
In particular, the crossratio of four spheres that are pairwise related by a Lie inversion is given by the following formula (cf. [1, §53]):
Lemma 2.6
Let \(a \in {\mathbb {P}}({\mathbb {R}}^{4,2}) {\setminus } {\mathbb {P}}({\mathcal {L}})\) and \(s_1, s_2 \in {\mathbb {P}}({\mathcal {L}})\) be two spheres that do not lie in the linear sphere complex \(a^\perp \), then
Discrete Legendre maps
Discrete surfaces in this paper will be represented by discrete Legendre maps from a connected quadrilateral cell complex \({\mathcal {G}}=({\mathcal {V}},{\mathcal {E}},{\mathcal {F}})\) of degree 4 to the space of contact elements \({\mathcal {Z}}\). The set of vertices (0cells), edges (1cells) and faces (2cells) of the cell complex \({\mathcal {G}}\) are denoted by \({\mathcal {V}}\), \({\mathcal {E}}\) and \({\mathcal {F}}\), respectively.
The directions in the cell complex will be labelled by upper indices (1) and (2) and we obtain two distinguished sets of edges \({\mathcal {E}}^{(1)} \overset{.}{\cup } {\mathcal {E}}^{(2)}={\mathcal {E}}\). A (1) resp. (2)coordinate ribbon is then the sequence of faces bounded by two adjacent (1) resp. (2)coordinate lines.
Definition 2.7
[4, 10] A discrete line congruence \(f: {\mathcal {V}} \rightarrow {\mathcal {Z}}, i \mapsto f_i\), is a discrete Legendre map if two adjacent contact elements \(f_i\) and \(f_j\) share a common curvature sphere \(s_{ij}:= f_i \cap f_j\).
Note that, for any discrete Legendre map, we therefore obtain two curvature sphere congruences \(s^{(1)}:{\mathcal {E}}^{(1)} \rightarrow {\mathbb {P}}({\mathcal {L}})\) and \(s^{(2)}:{\mathcal {E}}^{(2)} \rightarrow {\mathbb {P}}({\mathcal {L}})\).
Moreover, for any fixed point sphere complex \({\mathfrak {p}} \in {\mathbb {R}}^{4,2}\), \(\langle {{\mathfrak {p}}, {\mathfrak {p}}}\rangle =1\), the point sphere congruence \(p_i:= f_i \cap p^\perp \) of a discrete Legendre map provides a circular net, that is, any four point spheres of an elementary quadrilateral are concircular.
Discrete Rcongruences
Classically, a smooth sphere congruence is called Ribaucour if the curvature lines of its two envelopes correspond. Representing discrete surfaces by discrete Legendre maps, the analogous discrete problem, was studied in [6]: discrete sphere congruences enveloped by at least two generic discrete Legendre maps are given by Qnets in the Lie quadric, i.e., sphere congruences \(q:{\mathcal {V}} \rightarrow {\mathbb {P}}({\mathcal {L}})\) such that the four spheres of any elementary quadrilateral are coplanar.
In this paper, we will exclude degenerate faces of a Qnet and will assume that the spheres of a quadrilateral are not in oriented contact:
Definition 3.1
A map \(r:{\mathcal {V}} \rightarrow {\mathbb {P}}({\mathcal {L}})\) is called a discrete Rcongruence if the four spheres of any elementary quadrilateral lie in a unique linear system.
As pointed out in [6, §5], discrete Rcongruences with spheres lying in linear systems of signature (2, 1) provide the natural discrete counterparts to smooth Ribaucour sphere congruences. Furthermore, since in this case any four spheres of a quadrilateral are curvature spheres of a Dupin cyclide, these Qnets often allow for elegant and straightforward geometric interpretations. Thus, in this work, we will often focus on these discrete (2, 1)Rcongruences.
Remark
In the smooth case, Blaschke gave the following characterization of smooth Ribaucour sphere congruences [1, §77]: a smooth sphere congruence \(r:U \rightarrow {\mathbb {P}}({\mathcal {L}})\) is a Ribaucour sphere congruence if and only if for any choice of coordinates (u, v) there exists a map of osculating complexes \(t:~U \rightarrow ~{\mathbb {P}}({\mathbb {R}}^{4,2})\) such that
Similar osculating complexes also exist at vertices of a discrete Rcongruence: by definition, the nine Rspheres of the four adjacent quadrilateral lie in a common linear sphere complex. This complex is spanned by the vertex sphere and its four neighbours.
Any face of a discrete Rcongruence induces two special Lie inversions that will turn out to be crucial in the construction of its envelopes (see Fig. 2, right):
Proposition 3.2
A discrete Rcongruence induces a unique map of linear sphere complexes determined by
such that for any quadrilateral (ijkl) the corresponding Lie inversions \(\sigma _{n^{(1)}}\) and \(\sigma _{n^{(2)}}\) satisfy
these induced Lie inversions will be denoted by
Proof
Let \(r_i, r_j, r_k\) and \(r_l\) be four spheres of an elementary quadrilateral of a discrete Rcongruence (see Fig. 2). Since the four spheres lie in a linear system and are therefore linearly dependent, we can choose homogeneous coordinates such that
Then, by Lemma 2.2 (i), the vectors
define two linear sphere complexes with the desired properties.
Since the choice of homogeneous coordinates in (3) is unique up to a common scaling of \({\mathfrak {r}}_i, {\mathfrak {r}}_j, {\mathfrak {r}}_k\) and \({\mathfrak {r}}_l\), the linear sphere complexes \(n^{(1)}\) and \(n^{(2)}\) are uniquely defined. \(\square \)
We emphasize that the choice of homogeneous coordinates in (3) is local and depends on the quadrilateral under consideration. However, for special discrete Rcongruences there exist global homogeneous coordinates inducing the Lie inversions \(\sigma ^{(1)}\) and \(\sigma ^{(2)}\). Such an example is provided by isothermic sphere congruences as discussed in Sect. 3.3.
Lemma 3.3
For any quadrilateral, the two linear sphere complexes determined by \(n^{(1)}\) and \(n^{(2)}\) are involutive.
Proof
Without loss of generality, for any quadrilateral we choose homogeneous coordinates as given in Eq. (4). Then
\(\square \)
Lemma 3.4
Let \((r_i, r_j, r_k, r_l)\) be four spheres of an elementary quadrilateral of a discrete Rcongruence.

(i)
\(cr(r_i, r_j, r_k, r_l)< 0\) if and only if \(n^{(1)}\) and \(n^{(2)}\) determine two elliptic or two hyperbolic linear sphere complexes.

(ii)
\(cr(r_i, r_j, r_k, r_l)> 0\) if and only if the linear sphere complexes determined by \(n^{(1)}\) and \(n^{(2)}\) are of different type, that is, one linear sphere complex is elliptic and the other one is hyperbolic.
Proof
Suppose that \((r_i, r_j, r_k, r_l)\) are four spheres of an elementary quadrilateral of a discrete Rcongruence and choose homogeneous coordinates such that \(0={\mathfrak {r}}_i{\mathfrak {r}}_j+{\mathfrak {r}}_k{\mathfrak {r}}_l\). Then, by Lemma 2.6, we obtain that
and therefore conclude that
The points \(n^{(1)}\) and \(n^{(2)}\) lie in the plane spanned by the spheres \({r}_i, {r}_j, {r}_k\) and \({r}_l\). Therefore, if the plane has signature (2, 1), the linear sphere complexes are elliptic; if the spheres lie in a (1, 2)plane, then the linear sphere complexes are hyperbolic.
Moreover, \(cr(r_i, r_j, r_k, r_l) > 0\) if and only if
that is, if and only if the two linear sphere complexes are of different type (see Fig. 3 for the Möbius geometric picture). \(\square \)
This proposition also includes the special case of four concircular point spheres. It is wellknown that a circular quadrilateral is embedded if and only if the crossratio is negative. In this case, the Lie inversions \(\sigma ^{(1)}\) and \(\sigma ^{(2)}\) become Möbius inversions (see Fig. 3).
With the Lie inversions at hand, we are now prepared to discuss the main objects of interest, namely the envelopes of a discrete Rcongruence:
Definition 3.5
A discrete Legendre map \(f:{\mathcal {V}}\rightarrow {\mathcal {Z}}\) is the envelope of a discrete Rcongruence \(r:{\mathcal {V}}\rightarrow {\mathbb {P}}({\mathcal {L}})\), if \(r_i \in f_i\) for all \(i \in {\mathcal {V}}\).
Envelopes of a discrete Rcongruence can be constructed from one prescribed initial contact element using the inversions defined in Proposition 3.2. Thus, suppose that \(f_i:=\text {span} \{ s_0, r_i \}\) is an arbitrary initial contact element at the vertex \(i \in {\mathcal {V}}\). Then, the contact elements
define an envelope for the face (ijkl) of the discrete Rcongruence r: firstly, observe that by Lemmas 2.4 and 3.3 , the contact elements \(\sigma ^{(1)}(f_k)\) and \(\sigma ^{(2)}(f_i)\) indeed coincide. Furthermore, from Proposition 3.2 and Lemma 2.2, we deduce that the contact elements envelop the discrete Rcongruence,
and two adjacent ones intersect.
Moreover, due to Lemma 2.2(iii), this construction uniquely extends to all vertices \({\mathcal {V}}\) of the quadrilateral cell complex.
Conversely, any envelope of a discrete Rcongruence arises in this way:
Lemma 3.6
Let \(f:{\mathcal {V}} \rightarrow {\mathcal {Z}}\) be a discrete Legendre map enveloping the discrete Rcongruence \(r:{\mathcal {V}}\rightarrow {\mathbb {P}}({\mathcal {L}})\), then for any quadrilateral (ijkl) we obtain that
Proof
Assume that f is an envelope, then we know that \(r_i \in f_i\) and \(r_j \in f_j\) (see Fig. 4 left for the labelling of a quadrilateral). Therefore, the Lie inversion \(\sigma ^{(1)}\) preserves the curvature sphere \(s_{ij}=f_i \cap f_j\).
Hence, for any sphere \(r_i + \lambda s_{ij}\) in the contact element \(f_i\), it follows that the sphere
is contained in the contact element \(f_j\). This completes the proof. \(\square \)
Thus, we have reproven the following existence result for envelopes of a discrete Rcongruence that was already given in [4, Theorem 3.37]:
Corollary and Definition 3.7
Any envelope of a discrete Rcongruence is uniquely determined by the choice of an initial contact element and, therefore, any discrete Rcongruence admits a 2parameter family of envelopes.
This family of envelopes is said to be the Ribaucour family of a discrete Rcongruence. If the discrete Rcongruence consists of faces with signature (2, 1), the family is also called a (2, 1)Ribaucour family.
For later reference we remark on some relations between the envelopes and the Lie inversions associated to the discrete Rcongruence (see Fig. 4, right):
Lemma 3.8
Let \(f:{\mathcal {V}} \rightarrow {\mathcal {Z}}\) be a discrete Legendre map enveloping the discrete Rcongruence \(r:{\mathcal {V}}\rightarrow {\mathbb {P}}({\mathcal {L}})\). Then, for any quadrilateral, the Lie inversions \(\sigma ^{(1)}\) and \(\sigma ^{(2)}\) swap the curvature spheres on opposite edges:
Möbius geometric point of view
In this subsection we briefly discuss the interplay between the construction of Ribaucour families in Lie sphere and Möbius geometry. Thus, we fix a point sphere complex \({\mathfrak {p}} \in {\mathbb {R}}^{4,2}\), \(\langle {{\mathfrak {p}},{\mathfrak {p}}}\rangle =1\), to distinguish a Möbius geometry modelled on \(\langle {{\mathfrak {p}}}\rangle ^\perp \).
Since a Ribaucour family consists of discrete Legendre maps, the choice of a point sphere complex reveals a 2parameter family of enveloping circular principal contact element nets. The underlying circular nets are called the point sphere envelopes in \(\langle {{\mathfrak {p}}}\rangle ^\perp \) (see Fig. 5).
Note that a Lie inversion \(\sigma _{{\mathfrak {a}}}\) is a Möbius transformation, that is, it maps point spheres onto point spheres, if and only if \(\langle {{\mathfrak {a}},{\mathfrak {p}}}\rangle =0\). Therefore, generically, the Lie inversions \(\sigma ^{(1)}\) and \(\sigma ^{(2)}\) that map adjacent enveloping contact elements onto each other, do not transport point spheres along the discrete Rcongruence.
However, for each edge there exists a Möbius transformation uniquely determined by data from the discrete Rcongruence that relates point spheres of adjacent contact elements in the Ribaucour family:
Proposition 3.9
Let \(r: {\mathcal {V}} \rightarrow {\mathbb {P}}({\mathcal {L}})\) be a discrete Rcongruence. Then the Lie inversions \(\sigma _m\) with respect to the linear sphere complexes determined by
are Möbius transformations and satisfy, for any edge (ij),
Moreover, those Möbius transformations transport the contact elements of envelopes along the discrete Rcongruence and map adjacent point spheres of the point sphere envelope onto each other.
Construction of discrete Rcongruences
Discrete Rcongruences can be constructed from two arbitrary initial curves of spheres defined on two intersecting coordinate lines \(I^{(1)}\) and \(I^{(2)}\) of a quadrilateral cell complex \({\mathcal {G}}\) by the following iteration:

fix two initial sphere curves \(c^{(1)}: {\mathcal {V}}^{(1)} \rightarrow {\mathbb {P}}({\mathcal {L}})\) and \(c^{(2)}: {\mathcal {V}}^{(2)} \rightarrow {\mathbb {P}}({\mathcal {L}})\) that intersect at one vertex

choose for each face (ijkl) that contains an edge (ij) of the curve \(c^{(1)}\) a Lie inversion that maps the prescribed spheres \(c_i\) and \(c_j\) onto each other; for each face this amounts to the choice of a \(\lambda \in {\mathbb {R}}\):
$$\begin{aligned} n^{(1)}_{(ijkl)}:={\mathfrak {c}}^{(1)}_i  \lambda {\mathfrak {c}}^{(1)}_j \end{aligned}$$ 
starting at a face (ijkl) with three prescribed spheres \(c^{(1)}_i\), \(c^{(1)}_j\) and \(c^{(2)}_l\), we complete the face by defining
$$\begin{aligned} {\mathfrak {r}}_k:=\sigma ^{(1)}_{(ijkl)}({\mathfrak {c}}^{(2)}_l) \end{aligned}$$ 
iteratively this procedure gives a coordinate ribbon of a discrete Rcongruence including the spheres of the curve \(c^{(1)}\) and two spheres of the curve \(c^{(2)}\)

to obtain the next coordinate ribbon we choose again suitable Lie inversions along the just constructed sphere curve and proceed as described above.
Conversely, given a nowhere umbilic discrete Legendre map \(f: {\mathcal {V}} \rightarrow {\mathbb {P}}({\mathbb {R}}^{4,2})\), that is, opposite curvature spheres do not coincide, then any choice of spheres in the contact elements along two intersecting coordinate lines uniquely determines a discrete Rcongruence.
This construction relies on the following simple observation and is another instance of the interplay between line congruences and Qnets as studied in [5, 21]:
Lemma 3.10
Given four contact elements \((f_i, f_j, f_k, f_l)\) of an elementary quadrilateral of a nowhere umbilic discrete Legendre map and three spheres \(r_i \in f_i\), \(r_j \in f_j\) and \(r_l \in f_l\), then there exists a unique sphere \(r_k \in f_k\) such that \((r_i, r_j, r_k, r_l)\) provides a quadrilateral of a discrete Rcongruence.
Proof
By Lemma 2.3, there exists a unique Lie inversion \(\sigma \) satisfying
Then, \({\mathfrak {r}}_k:=\sigma ({\mathfrak {r}}_j) \in \text {span} \{ {\mathfrak {s}}_{jk}, {\mathfrak {s}}_{kl} \}\) provides the soughtafter sphere that completes the elementary quadrilateral of a discrete Rcongruence. \(\square \)
Special discrete Rcongruences
In this subsection we take up some recent ideas on discrete nets and special discrete Ribaucour transformations and briefly sketch how they fit into the framework developed in this work.
Circular nets. It is a wellknown fact that for any circular net there exists a 2parameter family of associated discrete Legendre maps.
We point out that a circular net can be understood as a special discrete Rcongruence, where all Rspheres, namely the point spheres, lie in the fixed hyperbolic linear sphere complex determined by the point sphere complex \(\langle {{\mathfrak {p}}}\rangle ^\perp \). Then the associated discrete Legendre maps provide the Ribaucour family of this sphere congruence.
Discrete Ribaucour coordinates. Discrete Rcongruences consisting of spheres that lie in a fixed parabolic or elliptic linear sphere complex \(\langle {{\mathfrak {a}}}\rangle ^\perp \) lead to particular Ribaucour families: for any choice of point sphere complex, there exists an envelope with point spheres lying on a sphere; namely, the sphere related to the fixed linear sphere complex \(\langle {{\mathfrak {a}}}\rangle ^\perp \). These envelopes are then discrete Ribaucour coordinates as discussed in [9, Theorem 3.9].
Discrete isothermic Rcongruences. A discrete Qnet in the Lie quadric \({\mathbb {P}}({\mathcal {L}})\) is called isothermic if any diagonal vertex star of the Qnet lies in a projective 3dimensional subspace of \({\mathbb {P}}({\mathbb {R}}^{4,2})\) that does not contain the four outer spheres of the vertex star. In particular, in the realm of discrete \(\Omega \)nets [7, 10], that is, discrete Legendre maps spanned by two isothermic sphere congruences, those Qnets are intensively used.
Their characterization in terms of the existence of a Moutard lift (see [4, 7, 10]), immediately reveals a relation to the framework considered in this work:
Definition 3.11
A lift \(\mu : {\mathcal {V}} \rightarrow {\mathcal {L}} \subset {\mathbb {R}}^{4,2}\) of a discrete Qnet \(s: {\mathcal {V}} \rightarrow {\mathbb {P}}({\mathcal {L}})\) is called a Moutard lift if opposite diagonals are parallel, that is,
where \(\delta \mu _{ik}:= \mu _k  \mu _i\) and \(\delta \mu _{jl}:= \mu _l  \mu _j\).
This special global choice of homogeneous coordinates of a Moutard lift gives rise to additional Lie inversions that diagonally swap the Rspheres on each quadrilateral:
Proposition 3.12
A lift \(\mu : {\mathcal {V}} \rightarrow {\mathcal {L}} \subset {\mathbb {R}}^{4,2}\) of a discrete Rcongruence \(r: {\mathcal {V}} \rightarrow {\mathbb {P}}({\mathcal {L}})\) is a Moutard lift if and only if for each quadrilateral the Lie inversion \(\sigma ^\delta \) with respect to the linear sphere complex \(\langle \delta \mu _{ik} \rangle ^\perp \) diagonally interchanges the Rspheres:
Proof
Suppose that \(\mu \) is a Moutard lift, then the linear sphere complex is determined by
and we therefore obtain
Conversely, if we have
it follows that \(\delta \mu _{ik} \  \ \delta \mu _{jl}\) and \(\mu \) is indeed a Moutard lift. \(\square \)
The interaction of these Lie inversions \(\sigma ^\delta \) induced by a Moutard lift with the geometric data of the envelopes is illustrated in Fig. 6.
Ribaucour families
Geometric properties of envelopes in the Ribaucour family
In the following paragraphs we discuss various special properties of the 2parameter family of envelopes in the Ribaucour family of a discrete Rcongruence.
Curvature spheres in a Ribaucour family
In Lemma 3.6 we saw that the Lie inversions \(\sigma ^{(1)}\) and \(\sigma ^{(2)}\) transport the contact elements of the envelopes along the discrete Rcongruence. Since two adjacent contact elements share a common sphere, namely the curvature sphere, this sphere is fixed by the corresponding Lie inversion. Therefore, we obtain the following property for all curvature spheres in a Ribaucour family:
Proposition 4.1
The curvature spheres of opposite edges of any envelope in a Ribaucour family lie in the same linear sphere complex.
Totally umbilic faces
If the four curvature spheres of a face of a discrete Legendre map coincide, the face is called umbilic. While in the smooth case umbilic points are rather special, in any (2, 1)Ribaucour family we obtain envelopes with umbilic faces:
Lemma 4.2
Let f be a discrete Legendre map in a (2, 1)Ribaucour family. A face (ijkl) of f is umbilic if and only if the four contact elements \(f_i, f_j, f_k\) and \(f_l\) are also contact elements of the Dupin cyclide generated by the Rspheres of this face.
In this case, for any projection to a Möbius geometry, the four point spheres of f lie on a curvature circle of the Dupin cyclide.
Proof
Let f be a discrete Legendre map in a (2, 1)Ribaucour family and assume that the contact elements \(f_i, f_j, f_k\) and \(f_l\) of a face (ijkl) coincide with contact elements of the Dupin cyclide \(d= D_1 \oplus _\perp D_2\), where the Rspheres \(r_i, r_j, r_k\) and \(r_l\) lie in the (2, 1)plane \(D_1\). Then, \(f_i=\text {span} \{ r_i, s_i \}\), where \(s_i \in D_2\). But, since \(s_i\) is also in oriented contact with the Rspheres \(r_j, r_k\) and \(r_l\), the sphere \(s_i\) is the constant curvature sphere of the envelope f at this face. Hence this face of f is umbilic.
Conversely, suppose that an envelope f of a discrete (2, 1)Rcongruence has a totally umbilic face, then the constant curvature sphere has to be in oriented contact with all four Rspheres of this face. Since these four Rspheres determine a Dupin cyclide, the constant curvature sphere of f is also a curvature sphere of this Dupin cyclide (in the other curvature sphere family than the Rspheres). This completes the proof. \(\square \)
Recall that, by Corollary 3.7, any choice of an initial contact element at an initial vertex provides a unique envelope in the Ribaucour family. Therefore, with the help of Lemma 4.2, we can construct umbilic faces at any face of a discrete (2, 1)Rcongruence:
Corollary 4.3
For any face of a discrete (2, 1)Rcongruence there exists a 1parameter family of envelopes in the Ribaucour family that are umbilic at this face.
Permutability theorems
A key property in the theory of transformations is the existence of permutability theorems: given two Ribaucour transforms \(f_1\) and \(f_2\) of a Legendre map f, then there exists a 1parameter family of Legendre maps that are simultaneous transforms of \(f_1\) and \(f_2\). Moreover, corresponding points of these Legendre maps are concircular. This result, generically, holds for smooth, as well as, for discrete Legendre maps (cf. [4, Theorem 3.6], [8, 22, §8]).
However, contrary to the smooth case, for any discrete Legendre map circularity of corresponding points in the permutability theorem can fail: four envelopes in a Ribaucour family obviously satisfy the permutability theorem. But, if we project to a Möbius geometry, corresponding point spheres of the four envelopes do not have to lie on a circle. The point spheres lie on the corresponding Rsphere and are therefore in general only cospherical.
Proposition 4.4
Let f be a discrete Legendre map, then there exist three Ribaucour transforms \(f_1\), \(f_2\) and \(f_{12}\) of f such that \(f_{12}\) is a simultaneous Ribaucour transform of \(f_1\) and \(f_2\) and, for any projection to a Möbius geometry, corresponding point spheres of these four nets are not circular.
We emphasize that if corresponding contact elements of f, \(f_1\), and \(f_2\) span a 3dimensional projective subspace of signature (2, 2), then there only exists a 1parameter family of simultaneous Ribaucour transforms \(f_{12}\). The points of the contact elements will then be circular. Hence in this case, the usual permutability theorem holds.
We remark that the significant difference to the smooth transformation theory pointed out in Proposition 4.4 leads to examples of discrete nets that have no counterparts in the smooth surface theory (see for example [24, §1.1]).
Deformations of envelopes in the Ribaucour family
The ambiguity of envelopes in the Ribaucour family of a discrete Rcongruence \(r:{\mathcal {V}} \rightarrow {\mathbb {P}}({\mathcal {L}})\) provides a possibility to smoothly deform two envelopes f and \({\hat{f}}\) into each other.
Let \(r_0\) be an Rsphere of the congruence r at an initial vertex \(v_0 \in {\mathcal {V}}\). Then, by Corollary 3.7, any choice of a smooth initial Legendre curve
gives rise to a 1parameter family \(\{f^{\gamma _0(t)}\}_{t \in [0,1] }\) of envelopes in the Ribaucour family (see Fig. 7).
In particular, this construction yields a smooth Legendre curve \(t \mapsto f^{\gamma _0(t)}_{v}\) for each vertex \(v \in {\mathcal {V}}\) that lies in the fixed parabolic linear sphere complex determined by the corresponding Rsphere \(r_v\). Moreover, according to Lemma 3.6, two adjacent Legendre curves are related by a Lie inversion. Thus, they envelop a 1parameter family of spheres and form a smooth Ribaucour pair of curves.
If projecting to a Möbius geometry, the point sphere envelopes of \(\{f^{\gamma _0(t)}\}_{t \in [0,1] }\) provide smooth deformations of circular nets, where the vertices move along spherical curves. In particular, if the initial curve \(\gamma _0\) is a (part of a) circle, then circularity of it is preserved along the entire discrete Rcongruence (cf. Proposition 3.9).
The construction pointed out in this section provides a way of obtaining discrete and semidiscrete triply orthogonal systems with special vertical coordinate surfaces having one family of spherical curvature lines. For particular choices of the initial curve \(\gamma _0\) we even obtain semidiscrete cyclic systems. A deeper analysis of these systems will be given in a future work.
Discrete Ribaucour pairs in the Ribaucour family
To gain further geometric insights into the Ribaucour family of a discrete Rcongruence, in this subsection, we fix two envelopes of a discrete Rcongruence, classically called a Ribaucour pair of discrete Legendre maps. The contact elements of a Ribaucour pair form a fundamental line system in the sense of [5, 21] since Rcongruences are Qnets.
Thus, suppose that \(f, {\hat{f}}:{\mathcal {V}}\rightarrow {\mathcal {Z}}\) are two discrete Legendre maps enveloping the discrete Rcongruence \(r:{\mathcal {V}}\rightarrow {\mathbb {P}}({\mathcal {L}})\). Then, for any edge (ij) the contact elements \((f_i, f_j, {\hat{f}}_j, {\hat{f}}_i)\) provide a quadrilateral of a discrete Legendre map, that is, adjacent contact elements share a common sphere. Thus, along each coordinate line of a Ribaucour pair the contact elements of f and \({\hat{f}}\) yield a coordinate ribbon of a discrete Legendre map; these coordinate ribbons will be called vertical ribbons of the Ribaucour pair. The “curvature spheres” of the vertical ribbons are given by the Rspheres and the curvature spheres of f and \({\hat{f}}\) along the corresponding coordinate line of the Ribaucour pair. Moreover, note that any two consecutive vertical ribbons also form a Ribaucour pair enveloping curvature spheres of f and \({\hat{f}}\).
In the following paragraphs we demonstrate how the structures of the various Ribaucour pairs interact with each other.
Proposition 4.5
Let \(f, {\hat{f}}:{\mathcal {V}}\rightarrow {\mathcal {Z}}\) be two envelopes of a discrete Rcongruence. Then there exists a map
such that for any quadrilateral (ijkl) the induced Lie inversion \(\sigma ^{(3)}:=\sigma _{n^{(3)}}\) interchanges the curvature spheres of f and \({\hat{f}}\),
Moreover, for any quadrilateral the Lie inversion \(\sigma ^{(3)}\) preserves the Rspheres, maps corresponding contact elements of the Ribaucour pair \((f, {\hat{f}})\) onto each other and is involutive to \(\sigma ^{(1)}\) and \(\sigma ^{(2)}\) (see Fig. 8 for notation).
Proof
Firstly observe that, since \(s_{kl}=\sigma ^{(2)}(s_{ij})\) and \({\hat{s}}_{kl}=\sigma ^{(2)}({\hat{s}}_{ij})\), we can choose homogeneous coordinates such that
as well as homogeneous coordinates
for the other four curvature spheres. Moreover, we define two vectors \(n^{(3)}\) and \({\tilde{n}}^{(3)}\) by
Then, the Rspheres \(r_i, r_j, r_k\) and \(r_l\) lie in the induced linear sphere complexes \(n^{(3)}\) and \({\tilde{n}}^{(3)}\), which are therefore involutive to \(n^{(1)}\) and \(n^{(2)}\).
Furthermore, since
the two linear sphere complexes induced by \(n^{(3)}\) and \({\tilde{n}}^{(3)}\) coincide:
\(\square \)
Thus, on each hexahedron of a Ribaucour pair \((f, {\hat{f}})\) we obtain the following symmetric configuration of the curvature spheres and the Rspheres:
In particular, for \(\nu =1,2,3\), the Lie inversion \(\sigma ^{(\nu )}\) fixes the four spheres assigned to the \((\nu )\)edges and interchanges the spheres within the other two linear sphere complexes.
Thus, the Lie inversions \(\sigma ^{(3)}\) given in Proposition 4.5 reveal a crucial property between two envelopes of a discrete Rcongruence:
Theorem 4.6
Two envelopes of a discrete Rcongruence are related by a facewise constant Lie inversion.
Furthermore, as a consequence of Lemma 2.4, the Lie inversions \(\sigma ^{(3)}\) constructed in Proposition 4.5 induce pairings in the entire Ribaucour family:
Corollary 4.7
Let \((f, {\hat{f}})\) be a discrete Ribaucour pair of a discrete Rcongruence. Then the corresponding Lie inversions \(\sigma ^{(3)}\) decompose the Ribaucour family into discrete Ribaucour pairs.
We emphasize that the Lie inversions \(\sigma ^{(3)}\) and the induced decomposition of the Ribaucour family depend on the initial choice of the Ribaucour pair \((f, {\hat{f}})\). Hence, the decomposition given in Corollary 4.7 is not unique.
Cyclidic nets in the Ribaucour family
For any face of a discrete Legendre map there exists a 1parameter family of facecyclides, Dupin cyclides that share the four curvature spheres assigned to a face with the discrete Legendre map (cf. [3, 24]). Note that any facecyclide has four distinguished curvature lines, namely the curvature lines that join two adjacent contact elements of the discrete Legendre map and lie on the corresponding curvature sphere. We will denote the space of (2, 1)planes in \({\mathbb {R}}^{4,2}\) by \(G_{(2,1)}({\mathbb {R}}^{4,2})\) and Dupin cyclides by two complementary (2, 1)planes \(D^{(1)}\) and \(D^{(2)}\) with \(D^{(1)} \oplus _\perp D^{(2)} = {\mathbb {R}}^{4,2}\).
To begin with, we shall point out how the Lie inversions \(\sigma ^{(1)}\) and \(\sigma ^{(2)}\) corresponding to a discrete Rcongruence interact with the facecyclides of its envelopes.
Lemma 4.8
A congruence of facecyclides of an envelope is preserved by the Lie inversions \(\sigma ^{(1)}\) and \(\sigma ^{(2)}\); however, the contact elements of opposite curvature lines going through the vertices of the discrete Legendre map are interchanged by the corresponding Lie inversion.
Proof
Let \(d=D^{(1)} \oplus _\perp D^{(2)}\) be a facecyclide of the face (ijkl) with \(s_{ij}, s_{kl} \in D^{(1)}\) and \(s_{il}, s_{jk} \in D^{(2)}\). Moreover, let \({\tilde{s}} \in D^{(1)}\) be a curvature sphere of the facecyclide. Then \(\langle {\tilde{{\mathfrak {s}}}, {\mathfrak {s}}_{il}}\rangle =\langle {\tilde{{\mathfrak {s}}}, {\mathfrak {s}}_{jk}}\rangle =0\) and we conclude that \({\tilde{s}}\perp n^{(1)}\). Thus, the spheres in \(D^{(1)}\) are fixed by the Lie inversion \(\sigma ^{(1)}\) and, therefore, also the facecyclide as unparametrized surface.
Furthermore, any contact element along the curvature line going through \(f_i\) and \(f_l\) is given by \(\langle {s_{il},{\tilde{s}}}\rangle \) for a sphere \({\tilde{s}}\in D^{(1)}\). Since
these contact elements are mapped to the contact elements of the opposite curvature line of the facecyclide. Analogous arguments for the other pair of curvature lines complete the proof. \(\square \)
This lemma also provides a construction for a special congruence
of facecyclides for f from a given facecyclide \(d_{\alpha }=D^{(1)}_{\alpha } \oplus _{\perp } D^{(2)}_{\alpha }\) of an initial face \((ijj'i')\) (for notations see Fig. 9): let
then \(d_\beta = D^{(1)}_{\beta } \oplus D^{(2)}_\beta \) is a facecyclide for the face \((jkk'j')\). Furthermore, defining
yields unique facecyclides \(d_\gamma \) and \(d_\delta \) for the four faces of the vertexstar. This construction consistently extends on all faces and provides a facecyclide congruence for the discrete Legendre map.
Projecting to any Möbius geometry \(\langle {\mathfrak {p}} \rangle ^\perp \), reveals a remarkable property of the just constructed facecyclide congruence d.
Firstly, observe that the facecyclides \(d_\alpha \) and \(d_\beta \) share a common curvature line \(c_{\alpha \beta }\), namely the circle of point spheres joining the point spheres \(p_j \in f_j\) and \(p_{j'} \in f_{j'}\). The contact elements along \(c_{\alpha \beta }\) coincide due to Eq. (7), and hence the two facecyclides define a piecewise smooth surface that is \(C^1\) across the common curvature line.
In the same way the pairs \(\{d_\beta , d_\gamma \}\) and \(\{d_\gamma , d_\delta \}\) meet at a common circular curvature line \(c_{\beta \gamma }\) and \(c_{\gamma \delta }\), respectively. Note that the circles \(c_{\alpha \beta }\) and \(c_{\beta \gamma }\), as well as \(c_{\beta \gamma }\) and \(c_{\gamma \delta }\), intersect orthogonally. Therefore, since the circular curvature lines \(c_{\alpha \beta }, c_{\beta \gamma }\) and \(c_{\gamma \delta }\) of the facecyclides around the vertexstar go through the common contact element \(f_{j'}\), also the facecyclides \(d_\delta \) and \(d_\alpha \) meet at a common curvature line \(c_{\delta \alpha }\). Since all contact elements of \(d_\delta \) and \(d_\alpha \) along the common circular curvature line \(c_{\delta \alpha }\) contain the curvature sphere \(s_{i'j'}\) of f, also the contact elements of \(d_\delta \) and \(d_\alpha \) along \(c_{\delta \alpha }\) coincide.
Thus, with the help of the Lie inversions \(\sigma ^{(1)}\) and \(\sigma ^{(2)}\), we have constructed a particular facecyclide congruence, discussed in [2, 3]:
Definition 4.9
Let \(f: {\mathcal {V}} \rightarrow {\mathcal {Z}}\) be a discrete Legendre map, then a cyclidic net of f is a congruence of facecyclides \(d: {\mathcal {F}} \rightarrow G_{(2,1)}({\mathbb {R}}^{4,2}) \times G_{(2,1)}({\mathbb {R}}^{4,2})\) such that two facecyclides adjacent along the edge (ij) share the same contact elements along the curvature direction going through \(f_i\) and \(f_j\).
The symmetries described in (6) can be exploited to relate cyclidic nets of a Ribaucour pair:
Theorem 4.10
Let \((f, {\hat{f}})\) be a discrete Ribaucour pair related by the facewise constant Lie inversion \(\sigma ^{(3)}\). If \(d=(D^{(1)}, D^{(2)}):{\mathcal {F}}\rightarrow G_{(2,1)}({\mathbb {R}}^{4,2}) \times G_{(2,1)}({\mathbb {R}}^{4,2})\) is a cyclidic net of f, then
provides a cyclidic net for \({\hat{f}}\).
Proof
Since corresponding curvature spheres of f and \({\hat{f}}\) are mapped onto each other by the Lie inversions \(\sigma ^{(3)}\) and Dupin cyclides are invariant under Lie inversions, \({\hat{d}}_{ijkl}\) defines a facecyclide for the face (ijkl) of \({\hat{f}}\).
Moreover, to prove that \({\hat{d}}\) indeed provides a cyclidic net of \({\hat{f}}\), we consider two adjacent faces along an edge (ij): firstly, observe that the Lie inversions \(\sigma ^{(3)}_{n}\) and \(\sigma ^{(3)}_{{\bar{n}}}\) belonging to the two adjacent faces are determined by the two linear sphere complexes
where \({\mathfrak {s}}_{ij} \in s_{ij}\), \(\hat{{\mathfrak {s}}}_{ij} \in {\hat{s}}_{ij}\) and \(\lambda , {\bar{\lambda }} \in {\mathbb {R}}\) are appropriately chosen. Thus, by Lemma 2.2(iii), the contact elements of the facecyclides of f along the common curvature line passing through \(f_i\) and \(f_j\) are mapped to the same contact elements by \(\sigma ^{(3)}_{n}\) and \(\sigma ^{(3)}_{{\bar{n}}}\). Thus, two adjacent facecyclides of \({\hat{d}}\) share common contact elements along the curvature line through \({\hat{f}}_i\) and \({\hat{f}}_j\). \(\square \)
In [27, Definition 4.4], the existence of two special Dupin cyclide congruences for a smooth Ribaucour pair of Legendre maps was pointed out. We report on a similar construction in the discrete case:
Definition 4.11
Let \(f, {\hat{f}}:{\mathcal {V}}\rightarrow {\mathcal {Z}}\) be two envelopes of a discrete Rcongruence. Facecyclides along a vertical ribbon will be called Rcyclides of the Ribaucour pair \((f, {\hat{f}})\), that is, for a vertical face, a Dupin cyclide \(\delta =R \oplus _\perp {\tilde{R}} \subseteq {\mathbb {R}}^{4,2}\) satisfying
In Theorem 4.10, we have learned that cyclidic nets for a Ribaucour pair arise in distinguished pairs, where the facecyclides are related by the Lie inversions \(\sigma ^{(3)}\). For these cyclidic nets there exists a canonical choice for the Rcyclides on the vertical ribbons:
Corollary 4.12
Suppose that d and \({\hat{d}}\) are cyclidic nets of a Ribaucour pair \((f, {\hat{f}})\) related by the Lie inversions \(\sigma ^{(3)}\). Then for an edge (ij) of the Ribaucour pair, the contact elements along two corresponding curvature lines of d and \({\hat{d}}\) passing through \(f_i\) and \(f_j\), as well as \({\hat{f}}_i\) and \({\hat{f}}_j\), uniquely determine an Rcyclide for the corresponding vertical face.
Proof
Since the contact elements along the curvature lines under consideration are related by the Lie inversion \(\sigma ^{(3)}\), two corresponding contact elements share a common sphere lying in \((n^{(3)})^\perp \). In this way, we obtain a 1parameter family of spheres that are in oriented contact with the spheres \(s_{ij}\) and \({\hat{s}}_{ij}\) and are therefore curvature spheres of a facecyclide for the vertical face. \(\square \)
We deduce that, by construction, the induced Rcyclides investigated in Corollary 4.12 provide (one ribbon of) a cyclidic net along each vertical ribbon. However, observe that two adjacent induced Rcyclides belonging to two vertical ribbons from different coordinate directions do not share a common curvature line. In particular, these Dupin cyclides do not give a 3D cyclidic net as introduced in [3, Section 3.2].
Envelopes with spherical curvature lines
In this section, we will draw attention to the Ribaucour transformation of discrete channel surfaces as discussed in [24] and the wider class of discrete Legendre maps with a family of spherical curvature lines.
Discrete spherical curvature lines
Inspired by the classification of spherical curvature lines in the smooth case (see [1, 14]), we introduce the notion of osculating complexes for discrete Legendre maps. To obtain uniqueness of the osculating complexes, we suppose a mild genericity condition on the discrete Legendre map. Note that also in the smooth case, uniqueness fails for the class of channel surfaces.
Thus, let \(f: {\mathcal {V}} \rightarrow {\mathcal {Z}}\) be a discrete Legendre map and fix a point sphere complex \({\mathfrak {p}}\in {\mathbb {R}}^{4,2}\), \(\langle {{\mathfrak {p}},{\mathfrak {p}}}\rangle < 0\). Furthermore, suppose that four consecutive contact elements \(f_{i'}, f_i, f_j\) and \(f_{j'}\) along a coordinate line are nowhere circular, that is, the four point spheres \(p_{i'}, p_i, p_j\) and \(p_{j'}\) do not lie on a circle.
Then the spheres of these four contact elements lie in a unique elliptic linear sphere complex: let s and \({\tilde{s}}\) be the two oriented spheres that contain the four point spheres \(p_{i'}, p_i, p_j\) and \(p_{j'}\). Then the soughtafter linear sphere complex is given by
where \(s_{ij}\) denotes the curvature sphere belonging to the edge (ij). Clearly, the spheres of the contact elements \(f_i:=\text {span} \{ s_{ij}, p_i \}\) and \(f_j:=\text {span} \{ s_{ij}, p_j \}\) lie in \(t^\perp \). Moreover, since \(s_{ii'}\in f_i\) and \(s_{jj'}\in f_j\), also the spheres of the contact elements
are contained in \(t^\perp \).
Definition 5.1
Let \(f:{\mathcal {V}} \rightarrow {\mathcal {Z}}\) be a nowhere circular discrete Legendre map, then
are called the osculating complexes of f.
As an immediate consequence of the above considerations, we can characterize spherical curvature lines of a discrete Legendre map:
Proposition 5.2
The (1)coordinate lines of a nowhere circular discrete Legendre map are spherical if and only if the osculating complexes along each (1)coordinate line are constant.
Ribaucour transformations of discrete channel surfaces
Curvature spheres of discrete channel surfaces are constant in the circular direction and hence the corresponding contact elements along the circular parameter lines lie in a parabolic linear sphere complex. We observe the following property if the spheres of an Rcongruence lie in a parabolic subcomplex:
Proposition 5.3
A discrete Rcongruence admits an envelope with constant curvature spheres along each (1)coordinate line if and only if along each (1)coordinate line the Rspheres lie in a parabolic complex and the map \(n^{(2)}\) is constant along each (1)coordinate ribbon.
Proof
Suppose that \(r:{\mathcal {V}}\rightarrow {\mathbb {P}}({\mathcal {L}})\) is a discrete Rcongruence admitting an envelope with a constant curvature sphere along each (1)coordinate line. Then, along each such coordinate line the constant curvature sphere and the Rspheres are in oriented contact and therefore lie in a fixed parabolic linear sphere complex.
Furthermore, let us consider a (1)coordinate ribbon bounded by two (1)coordinate lines \(\gamma _{i}\) and \(\gamma _{i+1}\). Moreover, without loss of generality, we choose homogeneous coordinates such that the induced Lie inversions \(\sigma ^{(2)}_\alpha \) and \(\sigma ^{(2)}_\beta \) of two faces adjacent along the edge (jk) of the coordinate ribbon are given by
where \(\lambda _\beta \in {\mathbb {R}}{\setminus } \{ 0 \}\) is a suitable constant.
Then, denoting the constant curvature spheres along the coordinate line \(\gamma _{i}\) by \(s^{(1)}_{i}\), we conclude that the spheres given by \(\sigma _{n_\alpha ^{(2)}}({\mathfrak {s}}^{(1)}_i)\) and \(\sigma _{n_\beta ^{(2)}}({\mathfrak {s}}^{(1)}_i)\) have to coincide. Hence, there exists a constant \(c \in {\mathbb {R}}{\setminus } \{ 0\}\) such that
Therefore, since \(r_j, r_k\) and \(s^{(1)}_{i}\) are linearly independent, each scalar factor has to vanish and we conclude that \(n^{(2)}_\alpha = n^{(2)}_\beta \). So it is constant along each (1)coordinate ribbon. Geometrically, the constant \(n^{(2)}\) is the intersection of the lines \(\langle {s^{(1)}_i, s^{(1)}_{i+1}}\rangle \) and \(\langle {r_j, r_k}\rangle \).
Conversely, assume that along each (1)coordinate line the Rspheres lie in a fixed parabolic complex. Then, in particular, along each (1)coordinate line all Rspheres are in oriented contact with the constant curvature sphere along this coordinate line. Furthermore, since the map \(n^{(2)}\) is constant along each (1)coordinate ribbon, the choice of an initial contact element containing the corresponding constant curvature sphere reveals the soughtafter envelope of the Rcongruence (cf. Lemma 3.6). \(\square \)
We recall that a discrete Legendre map is a discrete channel surface in the sense of HertrichJeromin et al. [24], if it admits a facecyclide congruence which is constant along one family of coordinate ribbons.
In particular, discrete channel surfaces can be characterized by special properties of their curvature sphere congruences [24, Proposition 2.4]: a discrete Legendre map is a discrete channel surface with circular (1)direction if and only if the curvature sphere congruence \(s^{(1)}\) is constant along any (1)coordinate line and the curvature spheres \(s^{(2)}\) determine a fixed (2, 1)plane along each (1)coordinate ribbon.
Theorem 5.4
A discrete Rcongruence is enveloped by two discrete channel surfaces with circular (1)direction if and only if the map \(n^{(2)}\) is constant along each (1)coordinate ribbon and the Rspheres along each (1)coordinate line are curvature spheres of a Dupin cyclide.
Proof
Let \(r:{\mathcal {V}}\rightarrow {\mathbb {P}}({\mathcal {L}})\) be a discrete Rcongruence enveloped by two discrete channel surfaces f and \({\hat{f}}\) with circular (1)direction. Then, by Proposition 5.3, the map \(n^{(2)}\) is constant along each (1)coordinate ribbon.
To prove the second property of the discrete Rcongruence, let us consider a (1)coordinate ribbon and denote the constant curvature spheres of f and \({\hat{f}}\) along the two boundary (1)coordinate lines by \(s_i, s_{i+1}, {\hat{s}}_i\) and \({\hat{s}}_{i+1}\), respectively. Furthermore, by HertrichJeromin et al. [24, Proposition 2.4], the other family of curvature spheres of f along this (1)coordinate ribbon lies in a (2, 1)plane \(D_i\).
Defining the projection onto \(\langle {{\hat{s}}_i}\rangle ^\perp \)
we deduce that all Rspheres along the (1)coordinate line lie in the (2, 1)plane \(\pi (D_i)\). Therefore, the Rspheres are curvature spheres of a Dupin cyclide.
Conversely, suppose that the Rspheres along a (1)coordinate line \(\gamma _{i_0}\) are curvature spheres of a Dupin cyclide, that is, they lie in a (2, 1)plane \(C_{i_0}\). Then, any choice of two initial contact elements \(f_0:=\text {span} \{ r_{i_0}, s_{i_0} \}\) and \({\hat{f}}_0:=\text {span} \{ r_{i_0}, {\hat{s}}_{i_0} \}\), where \(s_{i_0}, {\hat{s}}_{i_0} \in C_{i_0}^\perp \) provides two enveloping discrete channel surfaces. \(\square \)
As an immediate consequence of the 1parameter freedom in the choice of the initial contact element \(f_0\) in the proof of Theorem 5.4, we obtain the following corollary:
Corollary 5.5
If a discrete Rcongruence admits two discrete channel surfaces as envelopes, then there exists a 1parameter family of enveloping discrete channel surfaces.
Furthermore, we remark that Theorem 5.4 reveals how the constructions given in Sect. 3.2 yield discrete Rcongruences admitting a 1parameter family of discrete channel surfaces in their Ribaucour families.
In particular, suppose that a discrete Rcongruence consists of point spheres and satisfies the conditions of Theorem 5.4. Geometrically, those discrete Rcongruences are provided by circular nets where the point spheres of one family of coordinate lines lie on circles such that two adjacent ones are related by a Möbius inversion. Then, by Corollary 5.5, there exists a 1parameter choice of contact elements such that the constructed principal net provides a discrete channel surface, that is, the principal net has indeed a constant curvature sphere along each coordinate line of one family (for details see also [24]).
Using the Rcyclides of a discrete Ribaucour pair given in Definition 4.11, we observe that the geometric structure of two enveloping discrete channel surfaces is also reflected in the geometry of the vertical faces of the Ribaucour pair:
Corollary 5.6
A Ribaucour pair consists of two discrete channel surfaces with circular (1)direction if and only if along each vertical (1)coordinate ribbon there exists a constant Rcyclide.
Proof
Suppose that a discrete Rcongruence is enveloped by two discrete channel surfaces f and \({\hat{f}}\) with circular (1)direction. Then the contact elements of each (1)vertical coordinate ribbon provide a coordinate ribbon of a discrete channel surface: according to Theorem 5.4, the curvature spheres along the vertical ribbon, namely the Rspheres, lie in a (2, 1)plane. Furthermore, the other curvature spheres of the vertical Legendre map, given by the curvature spheres of f and \({\hat{f}}\), are constant (cf. [24, Proposition 2.4])
Hence, since the (1)vertical Legendre maps are discrete channel surfaces, there exists a constant facecyclide along each of these coordinate ribbons, which is then by definition also an Rcyclide of the Ribaucour pair \((f, {\hat{f}})\).
Conversely, if along each (1)vertical ribbon there exists a constant Rcyclide, the Rspheres along each ribbon lie in a fixed (2, 1)plane and, by Proposition 5.3, the map \(n^{(1)}\) is constant along each (1)coordinate ribbon of the Ribaucour pair. Thus, the claim follows from Theorem 5.4. \(\square \)
To conclude this section we remark on a general property of Ribaucour transforms of discrete channel surfaces. This also gives insights into the geometry of the other envelopes of a Ribaucour family containing a 1parameter family of discrete channel surfaces.
Proposition 5.7
The Ribaucour transforms of a discrete channel surface have a family of discrete spherical curvature lines.
Proof
Let f be a discrete channel surface with circular (1)direction and denote by r a discrete Rcongruence of f. Contemplate a (1)coordinate line with an adjacent coordinate ribbon: we denote by \(s^{(1)}_i\) the constant curvature sphere of f and by \(D^{(2)}_{ij}\) the (2, 1)plane containing the curvature spheres of the other curvature sphere congruence along the coordinate ribbon. Then, the contact elements of f along this coordinate line lie in the 3dimensional space \(s^{(1)}_i \oplus D^{(2)}_{ij}\). Hence, in particular, the Rspheres of the discrete Rcongruence along this coordinate line, as well as the elements \(n^{(1)}\) along the coordinate ribbon, lie in this space.
Therefore, the contact elements along each (1)coordinate line of any envelope of r lie in a fixed linear sphere complex and, by Proposition 5.2, we indeed obtain an envelope with a family of spherical curvature lines. \(\square \)
Notes
[1, §53]: Lineare Kugelscharen und Kugelkomplexe
References
Blaschke, W.: Vorlesungen über Differentialgeometrie III. Springer Grundlehren XXIX, Berlin (1929)
Bo, P., Pottmann, H., Kilian, M., Wang, W., Wallner, J.: Circular arc structures. ACM Trans. Graph. 30:#101, 1–11 (2011). Proc. SIGGRAPH
Bobenko, A., HuhnenVenedey, E.: Curvature line parametrized surfaces and orthogonal coordinate systems: discretization with Dupin cyclides. Geom. Dedic. 159, 207–237 (2012)
Bobenko, A., Suris, Y.: Discrete Differential Geometry: Integrable Structure, Graduate Studies in Mathematics, vol. 98. The American Mathematical Society, Providence (2008)
Bobenko, A.I., Schief, W.K.: Discrete line complexes and integrable evolution of minors. Proc. R. Soc. A: Math., Phys. Eng. Sci. 471(2175), 20140819 (2015)
Bobenko, A.I., Suris, Y.B.: On organizing principles of discrete differential geometry. Geometry of spheres. Russ. Math. Surv. 62(1), 1–43 (2007)
Burstall, F., Cho, J., HertrichJeromin, U., Pember, M., Rossman, W.: Discrete \(\Omega \)nets and Guichard nets (2020). ArXiv eprint ArXiv:2008.01447
Burstall, F., HertrichJeromin, U.: The Ribaucour transformation in Lie sphere geometry. Differ. Geom. Appl. 24(5), 503–520 (2006)
Burstall, F., HertrichJeromin, U., Lara Miro, M.: Ribaucour coordinates. Beitr. Algebra Geom. 60(1), 39–55 (2019)
Burstall, F., HertrichJeromin, U., Rossman, W.: Discrete linear Weingarten surfaces. Nagoya Math. J. 231, 55–88 (2018)
Burstall, F., HertrichJeromin, U., Rossman, W., Santos, S.: Discrete surfaces of constant mean curvature. RIMS Kokyuroku 1880, 113–179 (2014)
Burstall, F., HertrichJeromin, U., Rossman, W., Santos, S.: Discrete special isothermic surfaces. Geom. Dedic. 174, 1–11 (2015)
Cecil, T.: Lie Sphere Geometry, With Applications to Submanifolds. Springer, New York (2008)
Cho, J., Pember, M., Szewieczek, G.: Constrained elastic curves and surfaces with spherical curvature lines (in preparation)
Corro, A.V., Tenenblat, K.: Ribaucour transformations revisited. Commun. Anal. Geom. 12(5), 1055–1082 (2004)
Dajczer, M., Florit, L.A., Tojeiro, R.: The vectorial Ribaucour transformation for submanifolds and applications. Trans. Am. Math. Soc. 359(10), 4977–4997 (2007)
Dajczer, M., Tojeiro, R.: An extension of the classical Ribaucour transformation. Proc. Lond. Math. Soc. (3) 85(1), 211–232 (2002)
Dajczer, M., Tojeiro, R.: Commuting Codazzi tensors and the Ribaucour transformation for submanifolds. Results Math. 44(3–4), 258–278 (2003)
Doliwa, A.: Quadratic reductions of quadrilateral lattices. J. Geom. Phys. 30(2), 169–186 (1999)
Doliwa, A.: The Ribaucour congruences of spheres within Lie sphere geometry. In: Bäcklund and Darboux transformations. The geometry of solitons, pp. 159–166. The American Mathematical Society, Providence (2001)
Doliwa, A., Santini, P.M., Mañas, M.: Transformations of quadrilateral lattices. J. Math. Phys. 41(2), 944–990 (2000)
HertrichJeromin, U.: Introduction to Möbius Differential Geometry. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2003)
HertrichJeromin, U., Hoffmann, T., Pinkall, U.: A discrete version of the Darboux transform for isothermic surfaces. In: Discrete Integrable Geometry and Physics, volume 16 of Oxford Lecture Ser. Math. Appl., pp. 59–81. Oxford University Press, New York (1999)
HertrichJeromin, U., Rossman, W., Szewieczek, G.: Discrete channel surfaces. Math. Z. 294, 747–767 (2020)
Konopelchenko, B.G., Schief, W.K.: Threedimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 454(1980), 3075–3104 (1998)
Konopelchenko, B.G., Schief, W.K.: On the unification of classical and novel integrable surfaces. I. Differential geometry. R. Soc. Lond. Proc. Math. Phys. Eng. Sci. 459(2029), 67–84 (2003)
Pember, M., Szewieczek, G.: Channel surfaces in Lie sphere geometry. Beitr. Algebra Geom. 59, 779–796 (2018)
Saji, K., Teramoto, K.: Behavior of principal curvatures of frontals near nonfront singular points and their application (2020). arXiv:2003.07256v1
Schief, W.K.: On the unification of classical and novel integrable surfaces. II. Difference geometry. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459(2030), 373–391 (2003)
Tenenblat, K., Wang, Q.: Ribaucour transformations for hypersurfaces in space forms. Ann. Global Anal. Geom. 29(2), 157–185 (2006)
Terng, C.L.: Geometric transformations and soliton equations. In: Handbook of Geometric Analysis, No. 2, Volume 13 of Adv. Lect. Math., pp. 301–358. International Press, Somerville (2010)
Acknowledgements
The authors would like to thank Fran Burstall for helpful discussions about isothermic Qnets. Moreover, financial support by JSPS GrantinAid (as part of the FY2017 JSPS Postdoctoral fellowship) and TU Wien (Hörbiger Award) is gratefully acknowledged. This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”. Moreover, the authors are very grateful to the referee for several comments and helpful suggestions regarding notations in the manuscript.
Funding
Open access funding provided by TU Wien (TUW).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Rörig, T., Szewieczek, G. The Ribaucour families of discrete Rcongruences. Geom Dedicata 214, 251–275 (2021). https://doi.org/10.1007/s10711021006141
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711021006141
Keywords
 Discrete differential geometry
 Lie sphere geometry
 Ribaucour transformation
 Discrete Legendre maps
 Lie inversions
Mathematics Subject Classification
 53A40
 53B25
 37K25
 37K35