# Ribaucour coordinates

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## Abstract

We discuss results for the Ribaucour transformation of curves or of higher dimensional smooth and discrete submanifolds. In particular, a result for the reduction of the ambient dimension of a submanifold is proved and the notion of Ribaucour coordinates is derived using a Bianchi permutability theorem. Further, we discuss smoothing of semi-discrete curvature line nets and an interpolation by Ribaucour transformations.

## Keywords

Channel surface Canal surface Semi-discrete surface Curvature line net Discrete principal net Circular net Ribaucour transformation Ribaucour coordinates## Mathematics Subject Classification

53C42 53A10 53A30 37K25 37K35## 1 Introduction

This paper touches upon several ideas and results concerning the Ribaucour transformation of submanifolds and, in particular, of curves: the 1-dimensional case not only helps to illustrate the main ideas of our investigations, but it is also of interest for the construction of 2-dimensional discrete and semi-discrete principal “circular” nets and for their “smoothing”, cf Burstall et al. (2016). On the other hand, our discussion of higher dimensional submanifolds not only provides a generalization of some of the results for curves, but it also sheds light on the reasons and structure behind the results for curves. Thus we first present several results for the Ribaucour transformation of curves and then discuss whether and how these generalize to higher dimensions: a reduction of the ambient dimension by means of Ribaucour transformation, which leads to Ribaucour coordinates; smoothing of a sequence of Ribaucour transforms; and the possibility of an interpolation between curves or submanifolds by means of a sequence of Ribaucour transformations. Each of these problems is addressed in both the smooth and discrete settings.

Ribaucour coordinates were used by the classical authors to investigate the geometry of surfaces with particular properties of their curvature lines: given a surface and a suitable plane, for example, a tangent plane of the surface, one may locally construct a regular map of the surface to the plane by means of touching 2-spheres. These “Ribaucour coordinates” then map the curvature line net of the surface to an orthogonal net in the plane. For example, it is advantageous to employ this type of coordinates to investigate and construct surfaces of constant mean curvature \(H=1\) in hyperbolic space, cf de Lima and Roitman (2002) or Hertrich-Jeromin (2003, §5.5.27). These classical Ribaucour coordinates were generalized in more recent work to hypersurfaces, see Corro and Tenenblat (2004, Cor 2.10), and to submanifolds with flat normal bundle, see Dajczer et al. (2007, Thm 1).

One principal aim of this paper is to provide a more direct geometric approach to these generalized Ribaucour coordinates, much in the spirit of the classical authors. This approach relies on a geometric method to reduce the ambient dimension of a curve or submanifold by means of a Ribaucour transformation into a hypersphere, see Corollary 2.5, Theorem 2.6 for (smooth and discrete) curves and Corollary 3.6, Corollary 3.7 for submanifolds. Once the dimension reduction is established, the higher codimension version of Ribaucour coordinates relies on a Bianchi permutability result, cf Dajczer et al. (2007, Thm 2) and Hertrich-Jeromin (2003, §8.5.8). In our setting, this permutability result is readily verified by elementary means, leading to Ribaucour coordinates for any smooth submanifold with flat normal bundle resp any discrete circular net of higher codimension, see Theorems 3.8 and 3.9.

Another observation that relies on rather similar ideas leads to a “smoothing” of a semi-discrete curvature line net, since such a net can be thought of as a sequence of Ribaucour transforms of one curve. If two *m*-dimensional submanifolds envelop an *m*-sphere congruence then this sphere congruence provides a metric and connection preserving isomorphism field between normal bundles, see Lemmas 2.1 and 3.4, cf Corro and Tenenblat (2004, Cor 2.9). As a consequence, any pair of curves that envelop a 1-parameter family of circles, that is, form a Ribaucour pair, are two curvature lines of a channel surface, see Corollary 2.4 or Fig. 3: this result can be thought of as the semi-discrete version of a “smoothing” result for discrete circular nets, see Bobenko and Huhnen-Venedy (2012) and Bo et al. (2011), where a discrete circular net is “smoothed” by fitting Dupin cyclide patches into its facets. The higher dimensional version Corollary 3.5 emphasizes that the construction only depends on an enveloped *m*-sphere congruence, not the fact that the two submanifolds form a Ribaucour pair: if they do, more structure can be obtained.

Finally, we prove that any two (smooth or discrete) curves can be transformed into each other by a sequence of Ribaucour transforms, see Corollary 2.9. In contrast to Cauchy problems for discrete nets, cf Bobenko and Hertrich-Jeromin (2001, Sect 4) or Hertrich-Jeromin (2003, §8.4.9), this is a mixed boundary and initial value problem and allows to construct discrete or semi-discrete principal nets with given boundary data. This result hinges again on the aforementioned dimension reduction, which allows us to reduce the problem to planar curves. For planar curves, the result is then proved by a simple geometric construction in the discrete case, and as a trivial consequence of Burstall and Hertrich-Jeromin (2006, Thm 3.4) in the smooth case, see Theorem 2.8. However, as the flatness of a certain vector bundle is required—which is trivial in the 1-dimensional case, but not in higher dimensions—our interpolation result does not generalize to higher dimensions.

Though we start by formulating our results for curves in Euclidean space in order to make the text more accessible, we quickly resort to sphere geometric methods: as the Ribaucour transformation is a Lie geometric notion, many arguments are more efficient and transparent in the Möbius or Lie geometric settings. Since these techniques are discussed in great detail in other works, we will only give few hints or details where required, and refer the reader to, for example, Hertrich-Jeromin (2003) as well as to our previous papers Burstall and Hertrich-Jeromin (2006) or Burstall et al. (2016).

## 2 Space curves

*Ribaucour pair*of curves, that is,

*x*and \({\hat{x}}\) share a common tangent circle

*c*(

*u*) at corresponding points

*x*(

*u*) and \({\hat{x}}(u)\), cf Burstall et al. (2016, Def 2.1) and Fig. 1. We say that the two curves are mutual

*Ribaucour transforms*of each other. For regularity we assume \(x(u)\ne {\hat{x}}(u)\) for all \(u\in I\). Analytically, this relation between the two curves can be encoded by the reality of their

*tangent cross ratio*,

*x*and \({\hat{x}}\). In detail, consider the orthogonal decomposition

*Euclidean lift*of a curve in \({\mathbb R}^3\) into the light cone obtained from the isometric embedding

*lifts*\(\xi \in \Gamma (\langle o+x+{1\over 2}(x,x)\,q\rangle )\) of

*x*and \({\hat{\xi }}\) of \({\hat{x}}\), in fact, the map

*m*and radius

*r*or a plane with unit normal

*n*through

*x*in \({\mathbb R}^3\) by (cf Hertrich-Jeromin 2003, Sect 1.1)

*Ribaucour pair*if endpoints of corresponding edges are concircular, that is, for adjacent \(i,j\in I\),

*s*(

*u*) that contain corresponding points

*x*(

*u*) and \({\hat{x}}(u)\) of both curves and have

*n*(

*u*) as a normal; aligning orientations this construction yields an isometric isomorphism of normal bundles for the curves of a Ribaucour pair. Indeed, employing Möbius geometric lifts

## Lemma 2.1

Suppose \(x,{\hat{x}}:I\rightarrow {\mathbb R}^3\) form a Ribaucour pair of curves; then the isomorphism (2.7) of normal bundles maps parallel normal fields *n* of *x* to parallel normal fields \({\hat{n}}\) of \({\hat{x}}\), that is, (2.7) intertwines normal connections of the two curves.

*n*is parallel if and only if its lift (2.5) is,

*x*and \({\hat{x}}\) and yields parallel normal fields for both curves,

## Lemma 2.2

Suppose that a curve \(s:I\rightarrow S^{3,1}\) of spheres touches two curves \(x,{\hat{x}}:I\rightarrow {\mathbb R}^3\) and yields parallel normal fields for these curves; then *x* and \({\hat{x}}\) form a Ribaucour pair of curves.

As a curve on a surface is a curvature line if and only if the Gauss map of the surface yields a parallel normal field along the curve we deduce the following theorem for the channel surface enveloping one of the curves of spheres discussed above, cf Pember and Szewieczek (2017, Sect 5):

## Corollary 2.3

Any two (non-circular) curvature lines of a channel surface form a Ribaucour pair of curves; given a Ribaucour pair of curves \(x,{\hat{x}}:I\rightarrow {\mathbb R}^3\) there is a 1-parameter family of channel surfaces that contain both curves as curvature lines.

The first claim follows directly from Lemma 2.2, as the 1-parameter family of spheres enveloped by a channel surface yields parallel normal fields along any two of its (non-circular) curvature lines; the second claim follows from Lemma 2.1, by using one of the sphere congruences (2.6) given by a parallel normal field *n* along *x* to obtain one of the sought-after channel surfaces.

Recall that a semi-discrete curvature line net can be thought of as a sequence of curves, where subsequent curves form Ribaucour pairs, cf Müller and Wallner (2013, Def 1.1) or Burstall et al. (2016, Sect 3). As the construction of Corollary 2.3 of a channel surface from a Ribaucour pair of curves involves the choice of a parallel normal field along one of the two curves, this construction may be iterated to “smoothen” a semi-discrete curvature line net, see Fig. 3. reminiscent of the smoothing of fully discrete curvature line nets by using Dupin cyclides, cf Bobenko and Huhnen-Venedy (2012):

## Corollary 2.4

Any semi-discrete curvature line net can be “smoothed” by a sequence of channel surfaces: it lies on a \(C^1\)-surface composed of channel surfaces that meet at the curves of the semi-discrete net and have the same tangent planes there.

This relation between Ribaucour pairs of curves and enveloped (hyper-)sphere congruences also yields an approach to reduce the ambient dimension of a curve by means of a Ribaucour transformation: given a curve \(x:I\rightarrow {\mathbb R}^3\) we may use a parallel normal field \(n:I\rightarrow S^2\) to construct a sphere congruence touching a given, fixed sphere and thereby producing a Ribaucour transform of *x* that is contained in this sphere—as the radial vector field of a fixed sphere is parallel along any curve in that sphere the claim is an immediate consequence of Lemma 2.2.

*x*can be determined explicitly, by algebra alone: if

*x*and a parallel normal field

*n*along

*x*, then

*s*and

*e*, that is, of the desired Ribaucour transform \({\hat{x}}\) of

*x*.

## Corollary 2.5

Given a (fixed) sphere, any curve \(x:I\rightarrow {\mathbb R}^3\) that does not meet the sphere can be Ribaucour transformed into the sphere by means of a parallel normal field *n* along *x*; once a parallel normal field is given, the construction of the transform is algebraic.

The first part of Corollary 2.5 works in a completely analogous way for discrete curves: given a sphere that the curve does not meet, a Ribaucour transform of the curve can be constructed by iteratively constructing second intersection points of the sphere with the circumcircles of the endpoints of an edge and an initial point for the edge on the sphere, cf Fig. 4:

## Theorem 2.6

Given a sphere, any discrete curve \(x:{\mathbb Z}\supset I\rightarrow {\mathbb R}^3\) that does not meet the sphere can be Ribaucour transformed onto the sphere; this construction depends on the choice of one initial point on the sphere.

At no point of the argument did we use that the ambient dimension of the curve *x* be 3, hence Corollary 2.5 and Theorem 2.6 hold in any dimension—in particular, also for planar curves: this leads to *Ribaucour coordinates* for a given curve, similar to the Ribaucour coordinates used by the classical geometers for surfaces, cf de Lima and Roitman (2002), and generalized for submanifolds in Dajczer et al. (2007, Thm 1). Namely, iterating the dimension reduction of Corollary 2.5 (or Theorem 2.6) we obtain a circular arc as a double Ribaucour transform of a space curve \(x:I\rightarrow {\mathbb R}^3\). Suitably aligning iterated transformations, see Theorems 3.8 and 3.9, the order of transformations is irrelevant and we obtain Ribaucour coordinates for the given curve:

## Corollary and Definition 2.7

Any curve \(x:I\rightarrow {\mathbb R}^3\) is locally obtained by two subsequent commuting Ribaucour transforms from a circular arc;

the coordinates for the curve obtained in this way are called *Ribaucour coordinates* of the curve.

*Legendre lift*of a curve \(x:I\rightarrow {\mathbb R}^2\) with unit normal field \(n:I\rightarrow S^1\) by

*b*that intersects \(\lambda _i\) non-trivially defines a common Ribaucour transform \({\hat{x}}:I\rightarrow {\mathbb R}^2\) of the two initial curves \(x_i\).

Thus we obtain the following theorem that holds for both, smooth as well as discrete curves:

## Theorem 2.8

Any two planar curves \(x_0,x_1:I\rightarrow {\mathbb R}^2\) admit, under mild regularity assumptions, a common Ribaucour transform \({\hat{x}}:I\rightarrow {\mathbb R}^2\); in the smooth and discrete cases there is a 1- resp 2-parameter family of such common Ribaucour transforms.

As a consequence any two (discrete or smooth) space curves span a discrete resp semi-discrete curvature line net: we can solve a mixed boundary and initial value problem for (semi-)discrete curvature line nets. More precisely:

## Corollary 2.9

Any two discrete or smooth space curves \(x_0,x_1:I\rightarrow {\mathbb R}^3\) form two boundary curves of a discrete resp semi-discrete curvature line net; generically, three interpolating Ribaucour transforms are required.

## 3 Submanifolds

After putting forward the key ideas of this work in the previous section we are now prepared to tackle the general case of submanifolds in \({\mathbb R}^n\) or \(S^n\). In the case of space curves, we based our presentation on two lemmas to then easily deduce the results we were interested in—however, while Lemma 2.1 is insensitive to dimensions, and a useful reformulation will be readily available, Lemma 2.2 crucially depends on the “correct” dimensions, and we will therefore need to use different arguments to prove the results that depend on it.

In fact, the notion of a Ribaucour pair resp transform is more intricate in higher dimensions: in addition to the fact that two submanifolds envelop a congruence of spheres of the right dimension, a generalization of the classical demand that “curvature lines correspond” on the two envelopes needs to be implemented, cf Hertrich-Jeromin (2003, Def 8.2.2) and Burstall and Hertrich-Jeromin (2006, Sect 4):

## Definition 3.1

*n*-sphere, where \(\mathcal{L}^{n+1}=\{y\in {\mathbb R}^{n+1,1}\,|\,(y,y)=0\}\), form a

*Ribaucour pair*if

- (i)
they envelop a congruence of

*m*-spheres; - (ii)
the (1, 1)-subbundle \(x\oplus {\hat{x}}\) in \({\mathbb R}^{n+1,1}\) is flat.

Note that Definition 3.1(ii) yields a weak version of the condition that the curvature directions of two hypersurfaces correspond. Also note the slight change of notation: working in the conformal *n*-sphere from the start, *x* and \({\hat{x}}\) no longer denote immersions into \({\mathbb R}^n\) but into \(S^n={\mathbb P}(\mathcal{L}^{n+1})\).

## Lemma 3.2

- (i)envelop an
*m*-sphere congruence if and only if \(x^{(1)}\oplus {\hat{x}}=x\oplus {\hat{x}}^{(1)}\), where we denote$$\begin{aligned} x^{(1)}|_u:=x(u)\oplus d_u\xi (T_u\Sigma ^m) \quad \textit{for}\; x=\langle \xi \rangle \,\textit{and}\, u\in \Sigma ^m; \end{aligned}$$(3.1) - (ii)
form a Ribaucour pair if and only if \(d{\hat{\xi }}\in \Omega ^1(x^{(1)})\) for a suitable lift \({\hat{\xi }}\in \Gamma ({\hat{x}})\) of \({\hat{x}}\).

Lemma 3.2(i) says that the *m*-spheres touching *x* and containing the points of \({\hat{x}}\) coincide with those that touch \({\hat{x}}\) and contain the corresponding points of *x*; Lemma 3.2(ii) implies Lemma 3.2(i) and says that \(d{\hat{\xi }}\perp x\), hence that there is a parallel section \({\hat{\xi }}\) of \(x\oplus {\hat{x}}\).

At some points it will be useful to consider Legendre lifts into Lie sphere geometry, cf Burstall and Hertrich-Jeromin (2006), however we will mostly stay in Möbius geometry as we aim to keep the notions of points and circles, in particular with a view to the discrete setting, cf Bobenko and Hertrich-Jeromin (2001) or Hertrich-Jeromin (2003, §8.3.16):

## Definition 3.3

Two discrete circular nets \(x,{\hat{x}}:{\mathbb Z}^m\supset \Sigma ^m\rightarrow S^n={\mathbb P}(\mathcal{L}^{n+1})\) in the conformal *n*-sphere form a *Ribaucour pair* if corresponding edges are *circular*, that is, have concircular endpoints.

*m*-sphere congruence then we obtain a natural identification

*x*is mapped to a normal of \({\hat{x}}\) by means of a hypersphere that contains the

*m*-sphere of the enveloped sphere congruence,

## Lemma 3.4

If two immersed submanifolds \(x,{\hat{x}}:\Sigma ^m\rightarrow S^n\) envelop an *m*-sphere congruence then there is a natural, connection preserving isometric isomorphism (3.2) between their (weightless) normal bundles \(x^{(1)\perp }/x\) and \({\hat{x}}^{(1)\perp }/{\hat{x}}\).

*m*-sphere congruence yields one for the other via a “parallel” enveloped hypersphere congruence

*s*, hence both envelopes form

*extended curvature leaves*of the envelope of the

*m*-parameter family of hyperspheres, in the sense that their tangent spaces are invariant subspaces for the shape operator of the hypersurface, since

*m*-parameter family of hyperspheres \(s:\Sigma ^m\rightarrow S^{n,1}\) is foliated by spherical curvature leaves

*x*and \({\hat{x}}\) complementary to the spherical curvature leaves

*c*are tangent to the

*m*-spheres

Summarizing, we obtain the following higher dimensional version of Corollary 2.3:

## Corollary 3.5

If \(x,{\hat{x}}:\Sigma ^m\rightarrow S^n\) envelop an *m*-sphere congruence, and if *x* has a parallel normal field, then *x* and \({\hat{x}}\) yield extended curvature leaves of a hypersurface that is obtained as the envelope of an *m*-parameter family of hyperspheres.

Conversely, given the envelope of an *m*-parameter family \(s:\Sigma ^m\rightarrow S^{n,1}\) of hyperspheres in \(S^n\), where \(m<n-1\), any two extended curvature leaves complementary to its spherical curvature leaves envelop a congruence of *m*-spheres.

This generalization of Corollary 2.3 shows that the construction of a channel surface from a Ribaucour pair of curves only hinges on the enveloped circle congruence and the existence of parallel normal fields, not on the fact that they form a “Ribaucour pair” in the more general sense: if we demand higher dimensional submanifolds *x* and \({\hat{x}}\) to form a Ribaucour pair in the sense of Definition 3.1 then more fine structure of *x* and \({\hat{x}}\) as extended curvature leaves of a hypersurface can be derived.

Note that the first construction of Corollary 3.5 can be iterated when a sequence of submanifolds that envelop *m*-sphere congruences in a suitable manner is given and if one of the submanifolds admits a parallel normal field: this yields a higher dimensional version of Corollary 2.4.

*x*to construct a congruence of hyperspheres

*x*and

*e*, that is, \(s\perp \xi ,d\xi \) and \((s,e)\equiv 1\). The touching points

*s*and

*e*then form a Ribaucour transform of

*x*, by Lemma 3.2, as soon as \(t+x\) is a parallel normal field of

*x*, since

*x*, that is, that

*x*does not intersect the hypersphere

*e*, we learn that this condition is also necessary:

## Corollary 3.6

*x*admits a parallel normal field \(t+x\in \Gamma (x^{(1)\perp }/x)\); in this case a spherical Ribaucour transform is given by

*x*and \({\hat{x}}\) form a Ribaucour pair as soon as they envelop an

*m*-sphere congruence: since \((ds,\xi )=-(s,d\xi )=0\) we conclude that

*given seven vertices of a cube with circular faces, the eighth vertex can be constructed uniquely*. Namely, considering a face of the original net

*x*and an initial point of the corresponding face of \({\hat{x}}\) the construction of the face of \({\hat{x}}\) is completed on the 2-sphere containing the five starting points, cf Bobenko and Hertrich-Jeromin (2001, Sect 4). A standard dimension count then shows that corresponding

*m*-cells of the circular nets

*x*and \({\hat{x}}\) lie on

*m*-spheres:

## Corollary 3.7

Any circular net \(x:{\mathbb Z}^m\supset \Sigma ^m\rightarrow S^n\) admits a Ribaucour transform to a fixed hypersphere \(e\in S^{n,1}\) that it does not meet, \(x\not \perp e\); the two nets *envelop* a discrete *m*-sphere congruence, in the sense that corresponding *m*-cells lie on *m*-spheres.

As in the case of curves in Corollary 2.7, the dimension reductions of Corollary 3.6 resp Corollary 3.7 can be used to introduce Ribaucour coordinates for a submanifold with flat normal bundle—where “enough” spherical Ribaucour transforms exist, cf Dajczer et al. (2007, Thm 1).

*m*-sphere in the conformal

*n*-sphere via

*x*into the

*m*-sphere

*e*: with a first Ribaucour transformation of

*x*into the

*i*th hypersphere \(e_i\) and its parallel normal field obtained by (3.2) from the

*j*th parallel normal field of

*x*,

*i*and

*j*, thus confirming Bianchi’s permutability theorem for the particular Ribaucour transformations we use, cf Burstall and Hertrich-Jeromin (2006, Sect 3) or Dajczer et al. (2007, Cor 16). Once the circularity claim of Bianchi’s permutability theorem is established for \(x,x_i,x_j\) and \(x_{ij}\),

*x*does not hit the hyperspheres \(e_i^\perp \) and \(e_j^\perp \), and \(x_i\) and \(x_j\) do not hit the target \((n-2)\)-sphere \(\langle e_i,e_j\rangle ^\perp \) of \(x_{ij}\): we require that the dimension reduction of Corollary 3.6 works in every step of the iteration.

Using (3.5) it is now straightforward to formulate the general case: as every permutation of a finite set is a composition of transpositions our dimension reduction is independent of order, and circularity of Bianchi quadrilaterals ensures that (3.5) generalizes. More precisely,

## Theorem and Definition 3.8

*m*-sphere, given in terms of orthogonally intersecting hyperspheres \(e_1,\dots ,e_{n-m}\in S^{n,1}\). Then \({\hat{\xi }}\) of the form

*x*into the

*m*-sphere

*e*.

We say that \({\hat{x}}\) defines *Ribaucour coordinates* for the submanifold *x*.

Note that the choices of (parallel resp constant) normal fields for *x* and the *m*-sphere *e* establishes an isometric and connection preserving isomorphism between normal bundles: this isomorphism is the source of our vectorial Ribaucour transformation that establishes the Ribaucour coordinates, cf Dajczer et al. (2007, Thm 1).

Based on Corollary 3.7 Ribaucour coordinates for discrete principal nets can be introduced in a similar way: in order to ensure that permutability holds we need to choose initial points for the spherical Ribaucour transforms suitably—Miguel’s theorem then guarantees that the double Ribaucour reduction \(x_{ij}\) is independent of order and corresponding points of the circular nets \(x,x_i,x_j\) and \(x_{ij}\) are concircular, see Hertrich-Jeromin (2003, Sect 8.5). Thus in the case of an *m*-dimensional net in \(S^n\) initial points need to be suitably chosen on an \((n-m-1)\)-sphere containing the initial point of *x* and the corresponding point of the target net defining the Ribaucour coordinates.

## Theorem and Definition 3.9

*m*-sphere \( e = \langle e_i\,|\,i=1,\dots ,m\rangle ^\perp \) in terms of orthogonally intersecting hyperspheres \(e_1,\dots ,e_{n-m}\in S^{n,1}\); further fix \(i_0\in \Sigma ^m\) and choose an \((n-m)\)-cube

*We say that* \(\hat{x}\) *defines* Ribaucour coordinates *for the discrete circular net* *x*.

While the dimension reduction via Ribaucour transformations works perfectly in higher dimensions, the interpolation by Ribaucour transformations of Theorem 2.8 hinges on the flatness of the Demoulin vector bundle (2.10)—which is no longer trivially satisfied in higher dimensions. Indeed, in Burstall and Hertrich-Jeromin (2006, Sect 6) we provide an example of two surfaces that have one common (totally umbilic) Ribaucour transform but not more, so that permutability fails: in particular, the curvature lines on the common Ribaucour transform (obtained from the respective Ribaucour transformation) do not line up.

This situation is typical: by introducing Ribaucour coordinates of Definition 3.8 for two *m*-dimensional submanifolds with flat normal bundle in \(S^n\), the two submanifolds may be (locally) obtained by a sequence of \(2(n-m)\) Ribaucour transformations from each other, however, their curvature directions will in general not line up in the desired way.

In the discrete case, where curvature line coordinates are inherent, we learn from these observations that interpolation by Ribaucour transformations fails in general: a simple discretization of the Dupin cyclides used in Burstall and Hertrich-Jeromin (2006, Sect 6), by sampling curvature lines, provides counter-examples.

## Notes

### Acknowledgements

Open access funding provided by TU Wien (TUW). This work would not have been possible without the valuable and enjoyable discussions with A. Fuchs, C. Müller, M. Pember, F. Rist and G. Szewieczek about the subject. Furthermore, we gratefully acknowledge financial support of the second author by the Bath Institute for Mathematical Innovation; and of the third author by the TU Wien Doctoral Training Centre “Computational Design”.

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