Isothermic triangulated surfaces
We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is Möbius invariant. Isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes. This definition generalizes isothermic quadrilateral meshes.
A consequence is a discrete analog of minimal surfaces. Here the Weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function.
Isothermic surfaces are central objects in classical Differential Geometry. They include all surfaces of revolution, quadrics, constant mean curvature surfaces and many other interesting surfaces . In particular, all classes of surfaces that are describable in terms of integrable systems in some way or other seem to be related to isothermic surfaces [12, 13].
A smooth surface in Euclidean space is called isothermic if it admits conformal curvature line parametrization around every point. Note however that there are various characterizations of isothermic surfaces that do not refer to special parametrizations.
Discrete Differential Geometry lies between Discrete Geometry and Differential Geometry. The geometry of a discrete surface is determined by the positions of a finite number of vertices, such as those of a triangulated surface in Euclidean space. Smooth surfaces in Differential Geometry can be regarded as limits of discrete surfaces by refinement. The goal of Discrete Differential Geometry is to look for mathematical structures on discrete surfaces as rich as their smooth counterparts. It has many applications, for example in computer graphics and architectural design .
The same spirit applied to complex analysis has led to two different definitions of conformality for planar triangular meshes. One of these two is the theory of circle patterns , where the conformal structure is defined by the intersection angles of neighboring circumcircles. It is motivated by Thurston’s circle packings as a discrete analog of holomorphic functions . Another version of discrete conformality is based on conformal equivalence of triangle meshes [29, 39], where the conformal structure is defined by the length cross ratios of neighboring triangles. Luo introduced this notion when studying a discrete Yamabe flow. Its relation to ideal hyperbolic polyhedra was investigated in .
Previous definitions of discrete isothermic surfaces were all based on quadrilateral meshes that provide a discrete version of conformal curvature line parametrizations of isothermic surface [2, 4, 8]. Inspired by discrete integrable systems , Bobenko and Pinkall  considered quadrilateral meshes with factorized real cross ratios, which led to further investigation of discrete minimal surfaces and constant mean curvature surfaces . Recently, the notion of curvature was introduced to discrete surfaces with vertex normals [6, 21].
Here we aim for a definition of isothermic triangulated surfaces which does not involve conformal curvature line parametrizations. It is motivated by a known (although not well-known) characterization that a smooth surface in Euclidean space is isothermic if and only if locally it admits a nontrivial infinitesimal isometric deformation preserving the mean curvature. The only reference that we could find is from Cieśliński et al. , stating that this theorem was known in the 19th century.
Infinitesimal isometric deformations of triangulated surfaces have been extensively studied since Cauchy’s rigidity theorem of convex polyhedral surfaces [42, 14]. An infinitesimal deformation of a triangulated surface in space is an assignment of velocity vectors to all the vertices. We can then calculate the change of edge lengths. An infinitesimal deformation is called isometric if the edge lengths are preserved.
Suppose we have a realization of a triangulated surface such that each face of spans an affine plane. Given an infinitesimal isometric deformation , each triangular face rotates with an angular velocity given by a certain vector . These vectors satisfy a compatibility condition on every interior edge :
where is the left face of and is the right face.
On the other hand, it is well-known that the integral of the mean curvature has a very canonical discrete analogue . Here we have defined the mean curvature associated to edge as
If we further demanded on every edge then the infinitesimal isometric deformation would be trivial, i.e. an infinitesimal Euclidean deformation. Hence we consider instead the change of the integrated mean curvature around vertices
We are now ready to define isothermic triangulated surfaces. The smooth counterpart of the following formulation for isothermic surfaces is given by Smyth .
A non-degenerate realization of an oriented triangulated surface, with or without boundary, is called isothermic if there exists a -valued dual 1-form , not identically zero, such that
Here and denote the set of interior oriented dual edges and the set of interior vertices of .
The following is an immediate consequence of our definition.
A strongly non-degenerate realization of a simply connected triangulated surface is isothermic if and only if there exists an infinitesimal isometric deformation that preserves the integrated mean curvature around vertices but is not induced from Euclidean transformations.
The class of isothermic triangulated surfaces is Möbius invariant.
For a non-degenerate realization of a closed genus- triangulated surface the space of infinitesimal conformal deformations is of dimension greater or equal to . The inequality is strict if and only if is isothermic.
Suppose is a non-degenerate realization of a simply connected triangulated surface. Then is isothermic if and only if there exists an infinitesimal deformation that preserves the intersection angles of neighboring circumcircles and neighboring circumspheres but is not induced from Möbius transformations.
In Section 6 we show that our definition generalizes isothermic quadrilateral surfaces : Subdividing any isothermic quadrilateral surface in an arbitrary way we obtain an isothermic triangulated surface.
In Sections 7, 8 and 9 we provide examples of isothermic triangulated surfaces that are not obtained via quadrilateral isothermic surfaces. Triangulated cylinders generated by discrete groups as well as certain planar triangular meshes and triangulated surfaces inscribed in a sphere are isothermic.
In Section 10 we introduce discrete minimal surfaces via a discrete analogue of the Christoffel duality. Our discrete minimal surfaces are obtained as the reciprocal-parallel meshes for triangulated surfaces (with boundary) inscribed in the unit sphere. This approach mirrors the property that a smooth minimal surface is a Christoffel dual of its Gauß map. The Weierstrass data needed to construct a discrete minimal surface consist of a planar triangular mesh and a discrete harmonic function. Such harmonic functions were first introduced by NcNeal . They were used in linear discrete complex analysis since Duffin  and have applications in statistical mechanics (see Smirnov ).
In Section 11, we review the smooth theory and prove some new theorems that are similar to discrete results established in earlier sections.
A triangulated surface is a finite simplicial complex whose underlying topological space is a connected 2-manifold with boundary. The set of vertices (0-cells), edges (1-cells) and triangles (2-cells) are denoted as , and .
Without further notice we assume that all triangulated surfaces under consideration are oriented.
A non-degenerate realization of a triangulated surface in is a map which is linear on each face and for every edge . We say is strongly non-degenerate if every face of spans an affine 2-plane.
We denote and the set of interior vertices and the set of interior edges respectively. We write as the oriented edge from the vertex to the vertex . Note that . The set of oriented edges is denoted by . The set of interior oriented edges is indicated by .
We recall some notions about discrete differential forms . A (primal) 1-form is a function defined on oriented edges of such that
A 1-form is closed if for every face
It is exact if there exists such that
It is easy to check that exactness implies closedness while the converse holds if the discrete surface is simply connected.
Similarly we consider a 1-form on the dual cell decomposition of and call a dual 1-form on . Here we denote the dual edge oriented from the right face of to the left face. The following notions are natural if we think of a dual 1-form on as a 1-form on . A dual 1-form is closed if for ever interior vertex
It is exact if there exists such that
where denotes the left face of and denotes the right face.
We distinguish dual 1-forms from primal 1-forms for the following reasons. Firstly, the closedness conditions are different. The closedness conditions are imposed on faces for primal 1-forms while they are imposed at vertices for dual 1-forms. Secondly, a discrete notion of the Hodge star operator is needed to identify 1-forms with dual 1-forms, although it is not explicitly used in this paper. In Discrete Exterior Calculus  one often uses the Hodge star operator, which maps a primal 1-form to a dual 1-form via
where denotes the angle of the triangle with respect to some discrete metric, i.e. an assignment of edge lengths. Given a dual 1-form and a primal 1-form , we will occasionally write
for some . Here we think of it as for some .
3. Möbius invariance
In this section we prove that the class of isothermic triangulated surfaces is invariant under Möbius transformations.
Given a triangulated surface and a Möbius transformation , we define as the triangulated surface with vertices . We consider only the Möbius transformations that do not map any vertex to infinity.
Taking to be minus the inversion in the unit sphere, we obtain a triangulated surface
Later we will identify with imaginary quaternions, which explains the notation . We are going to show that is isothermic if and only if is isothermic. We first rewrite the equations from Definition 1.1.
Given a non-degenerate realization of a triangulated surface, a -valued dual 1-form satisfies
if and only if there exists such that
Suppose satisfies for every interior vertex
Then, we have the identity
Using this it is easy to verify all our claims. ∎
With the above lemma, we can show is isothermic if and only if is isothermic.
Suppose a non-degenerate realization of a triangulated surface is isothermic with a non-trivial dual 1-form satisfying Definition 1.1. We write
for some . Then, the triangulated surface is isothermic with corresponding dual 1-form
Proof of Theorem 1.3.
It follows from the previous lemma and the fact that Möbius transformations are generated by inversions and Euclidean transformations. ∎
The above calculation can be simplified if written in terms of quaternions. Identifying the Euclidean 3-space with the space of purely imaginary quaternions we obtain
These two formulas are similar to the smooth case .
Lemma 3.1 provides another characterization of isothermic triangulated surfaces. We consider the light cone
Suppose is a non-degenerate realization of a triangulated surface and is a function. Then is isothermic with corresponding dual 1-form defined by
if and only if for every interior vertex
where is the lift of to defined by
A function satisfying Equation (5) is called a self-stress of .
It is known that the Möbius geometry of is a subgeometry of the projective geometry of . Möbius transformations of are represented as projective transformations of preserving the quadric defined by the light cone . If two non-degenerate realizations are related by a projective transformation, then the spaces of self-stresses of the two realizations are isomorphic . Hence, we obtain another proof of Theorem 1.3.
4. Infinitesimal conformal deformations
We consider infinitesimal conformal deformations for a given closed triangulated surface in space. We show that a surface is isothermic if and only if it is a singular point in the space of all surfaces conformally equivalent to the original one.
4.1. Conformal equivalence of triangle meshes
We recall that a discrete metric of a triangulated surface is a function satisfying the triangle inequality on every face. A non-degenerate realization induces a discrete metric via
Definition 4.1 ().
Two discrete metrics on a triangulated surface are conformally equivalent if there exists such that for every edge
Two non-degenerate realizations are conformally equivalent if their induced discrete metrics are conformally equivalent.
It leads naturally to an infinitesimal version of conformal deformations.
An infinitesimal deformation of a non-degenerate triangulated surface is a map . It is conformal if there exists such that the change of the induced discrete metric satisfies for every edge
In particular, is an infinitesimal isometric deformation if .
The conformal equivalence class of a triangulated surface in Euclidean space is Möbius invariant . It can be distinguished via logarithmic length cross ratios.
Given a discrete metric on a triangulated surface, its logarithmic length cross ratio is defined by
where is the left face of and is the right face.
Theorem 4.4 ().
Two discrete metrics and on a triangulated surface are conformally equivalent if and only if
Corollary 4.5 ().
The dimension of the space of the conformal equivalence classes of a triangulated surface is .
4.2. Infinitesimal deformations
In this section, we consider closed triangulated surfaces. Suppose is a discrete metric on a closed triangulated surface. We consider an infinitesimal change of the discrete metric and write it as for some infinitesimal scaling . Then the change of logarithmic length cross ratio on edge is given by
The image of the linear map is the tangent space of the space of conformal equivalence classes (which is the same space at all discrete metrics).
Given a closed triangulated surface. The operator is skew adjoint with respect to the standard product on given by for any .
Let be the function defined by on edge and zero on other edges. Then for any , we have
Thus we have . ∎
The above lemma implies that we have an orthogonal decomposition
Given a closed triangulated surface. We have the following.
It is obvious that
Assume . For each face we define
Suppose is the neighboring triangle sharing the edge with . Because of we have
Since the link of each vertex is a disk (although we only need the vertex link to be a fan), Equation (6) in fact defines a function such that for any edge
On the other hand, it is obvious that
the two vector spaces are indeed the same. ∎
Recall that conformal equivalence classes of a triangular mesh are parametrized by logarithmic length cross ratios. By the inverse function theorem the result below implies that by deforming a non-isothermic surface in space we can reach all nearby conformal equivalence classes. It is precisely in the case of an isothermic surface that the hypothesis of the inverse function theorem fails to be satisfied. Thus the space of all non-isothermic non-degenerate realizations in a fixed conformal equivalence class is a smooth manifold.
Suppose is a non-degenerate realization of a closed triangulated surface. Then is isothermic if and only if there exists a non-trivial element such that
for all infinitesimal scalings coming from infinitesimal extrinsic deformations in Euclidean space, i.e. for which there exists such that .
Suppose is isothermic with satisfying Definition 1.1. Let be an arbitrary infinitesimal deformation and we write . Since is closed, i.e. we have
From we obtain
we see that is well defined. Since we know that for every interior vertex
we thus have . Hence there exists an non-trivial element such that for every edge
Because is arbitrary we conclude that
for all infinitesimal scaling coming from infinitesimal extrinsic deformations.
On the other hand, suppose there exists a non-trivial such that
for all infinitesimal scaling coming from infinitesimal extrinsic deformations. We define a dual 1-form via
for every edge . Since , we have
In addition, for any infinitesimal deformation we write for some and . We obtain
Since is arbitrary we conclude that is closed, i.e.
Hence, is isothermic with dual 1-form . ∎
Proof of Theorem 1.4.
Consider the composition of maps
The space of infinitesimal conformal deformations is exactly . Moreover, we know
Finally we conclude: The inequality is strict is not surjective is isothermic. ∎
Since the conformal equivalence classes of a triangle mesh are parametrized by length cross ratios, we can rephrase the previous theorems as follows.
Given a closed triangulated surface, isothermic realizations are precisely the points in the space of all non-degenerate realizations where the map that takes a non-degenerate realization to the conformal equivalence class of its induced metric fails to be a submersion.
It is interesting to see how combinatorics affect geometry. It is known that the number of vertices of a closed genus- triangulated surface satisfies the Heawood bound 
This condition is known to be sufficient for the existence of a genus-g triangulated surfaces with vertices except for . Comparing the Heawood bound with the inequality in Theorem 1.4 we obtain more examples of isothermic surfaces.
Every non-degenerate realization of a closed triangulated surface with is isothermic.
The space of infinitesimal conformal deformations contains all deformations that come from infinitesimal Möbius transformations. Therefore this space has dimension at least 10 and hence a surface must be isothermic if . ∎
Some of these surfaces with small number of vertices can be realized in Euclidean space without self-intersection. For example, there are embedded surfaces with and as shown in .
5. Preserving intersection angles
Given a triangulated surface in Euclidean space, every triangle determines a circumscribed circle and every two triangles sharing an edge determines a circumscribed sphere if the vertices are not con-circular. Two circumscribed circles are called neighboring if their corresponding triangles share an edge. We will call two circumscribed spheres neighboring if they have a common vertex.
Intersection angles of circles and spheres are Möbius invariant. The intersection angles of neighboring circumcircles of a triangulated surface were used to define discrete Willmore functional .
Proof of Theorem 1.5.
Suppose we have an infinitesimal deformation that preserves the angles between circumcircles and circumspheres but is not induced from Möbius transformations. Then it cannot be that also preserves the length cross ratios (because it is not hard to see that in this case is an infinitesimal Möbius transformation). We write for some and . Then the change of logarithmic length cross ratios is where
(See Figure 4.) By our assumptions does not vanish identically.
We define a dual -form
Then we have
In order to show that is isothermic, we need to verify the closedness of , i.e. for every interior vertex i
We identify Euclidean space with the space of imaginary quaternions (Section 11). Pick any vertex and denote its neighboring vertices by . Then we take an inversion in the unit sphere centered at and denote the images of the neighboring vertices by . We have the following relations:
We define the infinitesimal scaling
By taking the logarithmic derivative of the following equation
we obtain for
where . On the other hand, the vertices form a closed polygon in . We define
Since the polygon is closed, we have
The fact that the deformation preserves the intersection angles of neighboring circles and neighboring spheres implies that the angles between the neighboring segments and osculating planes of the closed polygon remain constant. Thus there exists a constant vector such that
To show that the converse is true one only has to reverse the previous argument. ∎
6. Example: Isothermic quadrilateral surfaces
We show that isothermic quadrilateral surfaces as defined by Bobenko and Pinkall  are isothermic under our definition (after an arbitrary subdivision into triangles). Isothermic quadrilateral surfaces are analogous to conformal curvature line parametrizations of smooth isothermic surfaces. They can be treated using the theory of integrable systems. New isothermic surfaces can be obtained from a given isothermic surface via the Christoffel duality and Darboux transformations . Special discrete surfaces related to isothermic quadrilateral meshes were studied in [1, 3].
We first review some results on isothermic quadrilateral surfaces from . Then we construct an infinitesimal isometric deformation for every isothermic quadrilateral surface and show that the change of mean curvature around each vertex is zero. In this way we obtain isothermic triangulated surfaces from the earlier notion of isothermic quadrilateral surfaces.
Definition 6.1 ().
A discrete isothermic net is a map , for which all elementary quadrilaterals have factorized real cross-ratios in the form
where does not depend on and not depend on .
Theorem 6.2 ().
Let be a discrete isothermic net. Then the discrete net defined (up to translation) by the equations
is isothermic. is called the Christoffel dual of .
We need a formula for the diagonals of its Christoffel dual (Corollary 4.33 in ).
Given a discrete isothermic net , the diagonals of any elementary quadrilateral of its Christoffel dual are given by
6.2. Infinitesimal flexibility of isothermic quadrilateral surfaces
Given a discrete isothermic net we first arbitrarily introduce a diagonal for each quadrilateral in order to get a triangulation. Then, we define infinitesimal rotations on faces as follows.
Suppose is an elementary quadrilateral of a discrete isothermic net and the diagonal is inserted. Then we get two triangles and . We define infinitesimal rotations and where and are the corresponding vertices of the Christoffel dual .