1. Introduction
A polyhedral surface is defined by stitching planar polygons along congruent edges. A polyhedral surface can be endowed with a conformal structure making it a Riemann surface [23, 8]. The celebrated uniformization theory [3, 10] then implies the existence of a conformal map between ( is typically 3 in applications) and a topologically equivalent domain in the plane . The main focus of this paper is building constructive approximations of this map for polyhedral surfaces with generally shaped polygonal faces. Topologically, we restrict our attention to disktype surfaces. To date, we are not aware of any other existing algorithm that is proved to converge in the limit to the uniformization map for general polyhedral surfaces (i.e., with polygonal faces of arbitrary shape).
The problem of approximating planar conformal mappings is considered wellunderstood and there is a wealth of methods that produce approximations to conformal mappings between planar domains [19, 18, 9]. Circlepacking [23], imitates conformal mappings by replacing infinitesimal circles with finite one, and was shown to converge to conformal mappings as the circles are refined [20, 24, 12]. However, it seems no full generalization to polyhedral surfaces with generally shaped faces exists [23]. Other constructions of “discrete uniformization” exist in the field of Discrete Differential Geometry (DDG) [5] where the focus is building a consistent and rich discrete theory. An example for such construction that can be used to compute discrete uniformization is by Springborn and coauthors [22, 4]. Discrete Ricci flow is another example [13]. Other popular constructions can be found in [16, 11, 21, 14, 7] however no proof of convergence to the uniformization map is provided for any of these methods sofar.
In this paper we construct algorithms for approximating the uniformization map with guarantees (i.e. with convergence proof). Without loosing any generality we can subdivide each polygon in into triangles and henceforth assume we have a triangulation , where , is the set of vertices; , the set of edges; and , the set of oriented planar faces (triangles).
We will construct successive approximations to over a series of regular subdivisions of the surface . By regular subdivision we mean that at each level we cut every face into four similar faces by connecting the midedges points, see inset figure, and Figure 1 (top row). Our approximations will be simplicial mappings of , namely piecewiseaffine (over faces) and continuous mappings into the complex plane , the collection of such mappings over will be denoted by . will converge locally uniformly to . We will restrict our attention here to topological disks, and take as a canonical domain the equilateral defined by its three corners . To set a unique target uniformization map we will mark three distinct (positively oriented) boundary vertices that will be mapped to the corners
(respectively). This fixes all the degrees of freedoms of the map. Figure
1 shows examples of simplicial approximations to for a series of three refinements of a particular polyhedral surface: the middle row shows planar checkerboard texture mapped by the inverse of the simplicial maps to visualize the “conformality” of the approximations. The bottom row shows the homeomorphic image of under onto the equilateral domain. Brightred color indicates high conformal distortion, while grey indicates low conformal distortion. Note that the approximations are improving as the mesh is refined. The main result on which we build upon when developing the algorithms in this paper is:Theorem 1.1.
Let be the uniformization map of a disktype polyhedral surface to the equilateral taking three prescribed boundary vertices of to the corners of . Let be the level subdivided version of .
If, for an arbitrary but fixed , the argument of is known up to an error of , then one can construct a series of nonempty convex spaces of simplicial maps of such that:

Every map is quasiconformal (QC) homeomorphism that maps onto , with some constant independent of .

Every series , where , converges locally uniformly to the uniformization map . That is, converges uniformly in any compact subset of to the identity map .
Let us clarify the assumption “the argument of is known upto an error of ”. What we mean by that is that at every point , we can choose (arbitrary) chart , , where , , and that we can point an angle in the range
for arbitrary small but fixed. Intuitively, if we know in which ”half” of ( denotes the unit circle) the argument of the derivative of the map we are looking for resides in, then we can approximate via a convex program, namely looking for an element in a known convex subset of the simplicial maps .
This theorem will be proved in several parts: first, in Section 3 we will show that contains atleast one simplicial map that is: 1) quasiconformal, namely a homeomorphism with bounded conformal distortion, 2) its conformal distortion is converging to 1 with almost linear rate. Second, in Section 4, we will show that any series of quasiconformal simplcial maps that their conformal distortion converge to 1 converges to the uniformization map as described in Theorem 1.1. In Section 5 we will study the space of quasiconformal simplicial maps and characterize a collection of convex subsets , . We will further show that given an approximation of the argument of , one can single out one of these convex subsets that will be nonempty. This will finish the proof of Theorem 1.1.
Building upon Theorem 1.1 we will suggest two algorithms. The first algorithm is exhaustive but theoretically fully justified: it will test many candidates , one of which is guaranteed to lead to a nonempty space . The number of candidates will be rather large but shown to be independent of (i.e., exponential in the number of original faces ). Due to the large number of candidates this algorithm will mainly have theoretical importance but only limited practical applicability. Nevertheless, as far as we are aware this is the first algorithm to approximate the uniformization map for general faced polyhedral surfaces.
The second algorithm will be greedy in nature: it will start with some arbitrary convex space , and will iteratively move to other convex spaces When terminating successfully it guarantees an approximation to the uniformization map, enjoying all the properties of the first algorithm (and Theorem 1.1). However, the drawback here is that we do not have a proof that it will always ends up with a nonempty space (i.e., terminate successfully). Nevertheless, it works well in practice and is much more computationally efficient than the (first) exhaustive algorithm.
We start by defining the smooth and discrete conformal structures of , set notations and a few preliminary lemmas.
2. Smooth and discrete conformal structures
As noted above, a polyhedral surface admits a smooth (classical) conformal structure. We start by defining it.
A conformal structure is defined by providing a conformal atlas, that is, a set of analytic coordinate charts [10]. We will define such an atlas already customized to our later constructions. We fix a constant , and for each vertex , denote by the set of indices of 1ring neighbors to vertex .
We distinguish three types of vertices in our surface : interior, boundary, and corner. Interior vertices are vertices in the interior of , boundary are on the boundary but not one of the three corners .
Interior charts
Let be an interior vertex. Set . Set the domain to be the interior of the convex hull (over the surface) of the points , where (see inset, where for one particular vertex is colored in green). We define the chart by first rigidly unfolding each triangle in onto the plane, taking to the origin, and second, composing each (now planar) triangle with the map , where is the sum of angles at vertex , where is the angle of face (adjacent to vertex ) at vertex . This composition is made (possibly by incorporating rigid transformations in the plane) such that is globally continuous, see [23, 8] for more details and the inset for an illustration. Let us denote .
Boundary charts
Let be a boundary vertex (). Set . The domain is defined as the interior of the convex hull of and , , and is defined via mapping the neighborhood congruently to the plane as before, composing it with , where .
Corner charts
For the three corner vertices we set and define , , similar to the boundary vertices’ charts with the exception of using the mapping , where .
All the transition maps are conformal and therefore define a conformal structure over :
Lemma 2.1.
The transition maps are holomorphic.
Proof.
The holomorphy of the transition maps can be understood (see e.g.,[8]) from the fact that these maps are compositions of similarities and the analytic maps (note that we avoid the origin) , . The holomorphy across edges can be verified using standard extension theorems of conformal maps. ∎
Now that we have a (smooth,classical) conformal structure, the notion of conformal mappings from to the plane is welldefined; a map is conformal if for every chart , the map defined over is conformal in the classical sense.
The main object of this paper is to approximate the conformal mapping that maps the polyhedral surface bijectively to the equilateral triangle . The existence and uniqueness of such a mapping is set by the uniformization theorem [3, 10]:
Theorem 2.2.
There exists a homeomorphism that is conformal in the interior of , ^{1}^{1}1 denotes the interior of the set .. The map is uniquely set once required to take to the corners of .
We will approximate by constructing quasiconformal simplicial mappings from subdivided versions of to .
The subdivided triangulations of are constructed by the standard subdivision rule in each triangle, where . For example, in the inset the red mesh shows , and we show in purple all the faces inside one face of .
We now turn to define the discrete conformal structure over . A discrete conformal structure is basically assigning angles to the corners of each triangle such that the angle sum of each triangle is . Equivalently, we can embed each triangle in the Euclidean plane and think of it upto a similarity transformation. A consistent discrete conformal structure will approximate the smooth one as the surface is refined. A simple way to do it is by mapping each triangle to the plane with the charts and taking its image Euclidean triangle to define its conformal structure. Obviously, using different atlas, or assignments of triangles to charts will lead to different discrete conformal structure, but as we will show, at the limit it won’t matter. So we will use the Atlas defined above, and describe an arbitrary assignment of faces to charts in .
We start by associating, for , each triangle to some chart , , by making sure that .
For example, we can use the assignment rule shown in the inset figure (each color indicates association of that colored face to a different vertex of the triangle and hence to a difference chart).
For any we associate the chart based on the triangle’s (unique) ancestor in level . We define the discrete conformal structure for each face by mapping the face’s vertices to the plane, that is , the associated chart , and using the Euclidean triangle to define the discrete conformal structure of . The inset below shows in blue (bottomleft) the triangle .
Once we defined the discrete conformal structure we can define the discrete conformal distortion of a simplicial map . We look per face , , and denote by the affine map mapping the triangle (remember that ) to the triangle . We will also refer to as expressed “in the local coordinate chart”. Then, the discrete conformal distortion of is defined by
where denotes the standard conformal distortion of the planar affine map , namely,
, the ratio of the larger to smaller singular values
of the linear part of the planar affine map . See the inset figure for an illustration.The discrete conformal distortion of the full simplicial map is accordingly defined by
(2.1) 
For later use, we denote by , , the discrete conformal distortion of the map restricted only to triangles contained in the set , that is
(2.2) 
Let us denote by the space of simplicial maps (i.e., continuous piecewise affine) mapping the triangulation to the plane, satisfying the boundary constraints of mapping ( denotes the boundary of the polyhedral surface ) bijectively to and taking the corner vertices to the triangle’s corners . Further define the subset , to include only orientationpreserving homeomorphisms such that . Denote , and by we will mean , that is, simplicial maps with maximum bound on its conformal distortion per face.
There are two technical issues to be taken careof for later constructions. First, let us highlight a small technical property of the charts and the association rule of faces we have defined that will be used in Section 3 when we approximate the uniformization map with simplicial map: we show that there exists some positive gap between the triangles and the boundary of the charts’ they are associated with. Denote the disk .
Lemma 2.3.
There exists some constant , such that
for all , associated with chart ,
Proof.
It is enough to prove the inclusion for . Take arbitrary face , and one of its vertices . Since when defining we took we see by the association procedure that every triangle is far from at least by some constant . When applying , since it has bounded derivatives (when considering it as map from the Euclidean faces to the complex plane) away from the vertex , we have some other bounding this distance from below. Taking the minimum over the (finite) set of faces in F and their vertices we finish the argument. ∎
A second issue is that we need to make sure that approximating the power maps , with simplicial maps (via sampling at the vertices and extending linearly) are quasiconformal with some universal bound on their distortion , independent of subdivision level . For an example of a simplicial power map see Figure 2.
We conjecture the following:
Conjecture 2.4.
Let be a triangle and denote the angle . Further let be the level of regular 14 subdivision of . Denote by the power map, and assume that . Then, the simplicial maps defined by sampling over the vertices of and extending by linearity are homeomorphisms that satisfy for some independent of .
For our needs it is enough to prove the following, slightly weaker, result:
Lemma 2.5.
Let be an isosceles triangle () and denote the angle . Further let be the level of regular 14 subdivision of . Denote by the power map, and assume that , and that . Then, the simplicial maps defined by sampling over the vertices of and extending by linearity are homeomorphisms that satisfy for some independent of .
The proof for this lemma is rather technical and therefore deferred to Appendix A.
Building upon this lemma, we can (without loosing generality) subdivide each of the original triangles of the polyhedral surface , so that the two conditions , and , where are satisfied. We can also guarantee that every triangle that touches one of the original vertices V is isosceles (all the other triangles have all flat vertex angles^{2}^{2}2By flat vertex angles we mean the angle sum around a vertex is . and so their charts are rigid congruencies and hence quasiconformal). Such subdivision is shown in the inset. Note that this subdivision produces a conformally equivalent polyhedral surface to with identity as the conformal equivalence, and therefore approximating the uniformization for this new polyhedral surface is equivalent to approximating the uniformization of the original polyhedral surface .
3. Feasibility
We are now ready to prove our first result. Namely, that a subdivided triangulation , , can be mapped to with a simplicial homeomorphism where the discrete conformal distortion of the triangles is controlled.
Theorem 3.1.
There exist simplicial maps , for , , where and face is associated with chart . That is, are orientation preserving homeomorphisms that satisfy for every face
In particular, , where , and .
Furthermore, for every face , , where is in local coordinates, is the centroid of the triangle , , , and are the vertices of the face .
Lastly, for any fixed domain satisfying , .
We will need an auxiliary Lemma regarding approximation of conformal mappings with simplicial maps:
Lemma 3.2.
Let be a conformal map such that . Let , , such that , and the minimal angle of the triangle is bounded from below, and denote . Then, the affine map , , defined uniquely by , , satisfies:

.

.

, with the constant inside the notation depending only upon the minimal angle of the triangle spanned by .

.
Proof.
We use the Taylor expansion (see for example,[2] page 179) of developed around :
(3.1) 
where is the circle of radius centered at the origin. We bound the reminder term for as follows. For :
Denote, for brevity . We can write three equations that characterize the affine map :
where . After rearranging:
Let us consider the matrix of the linear system:
Cramer’s rule implies that we can bound and by bounding , , where is identical to except that we replace its
column with the vector
. A direct calculation shows thatTherefore, , where denotes the area of the triangle spanned by . The terms of the form can be bounded by
Combining the above we get
where in the last equality we used the fact that the minimal angle in triangle is bounded from below and therefore . And similar bound holds for .
Since , there exists some constant depending on such that, upto an integer multiplication of , the difference are of the same order as , namely,
Finally,
where we used the fact that to bound away from zero. The lemma is proved. ∎
Proof.
(of Theorem 3.1) We construct by sampling the uniformization map , that is, for , we set , and extend to the whole by requiring linearity in each face.
Each face , , is associated with some chart , , (used to define its conformal structure, see Section 2). We denote, as above, by the (ordered) vertices of , and their image in the local coordinates.
We start with considering that are associated with interior charts, namely, charts for . Lemma 2.3 assures existence of a constant such that . Let us denote by the centroid of the triangle . Let us note that the edges’ length of are asymptotically of order . In the local coordinates (i.e., ) the edge length goes to zero not slower than , where (i.e., depends upon the vertex angles and how different they are from  angle deficit). Let us denote the radius of disk around containing , and the conformal map . The derivative of the conformal map, , can be bounded from below over , namely , (since , and therefore can be extended to a neighborhood of ). Since we have a finite number of charts and faces in F we can take to be a uniform lower bound for all charts.
We now would like to use Lemma 3.2. We assume w.l.o.g. that . We set , and note that . Also note that all the faces in every subdivision level are similar to the original faces F of . Lemma 2.5 therefore implies that the triangle have bounded angles from below. Lemma 3.2 now implies that the conformal distortion of the unique affine map taking to the triangle in spanned by is bounded by , since . In addition this lemma indicates that , where (asusual, the arguments are considered upto addition of ).
If we fix a domain such that then for the edge length of is asymptotically and we achieve that the conformal distortion of is bounded with . Lastly, for this case, note that Lemma 3.2 also indicates that for sufficiently high the affine map is orientation preserving and nondegenerate. Indeed since one can use Lemma 3.2 to show that for sufficiently large , .
Now we move to faces associated with boundary charts.
In this case we can use Schwarz reflection principle to extend to the union of and its reflection over its straight line boundary. Note that we can bound the derivative of the extension away from zero also in these cases (find a new that works also for boundary charts). Now we are again in the situation where we can apply Lemma 3.2. (we can adjust the constant to also work for these charts.)
The last case includes faces associated with corner charts , . This case can also be dealt with the Schwarz reflection principle, as follows. By definition of our conformal structure, the uniformization map in the corner coordinate charts , , takes the domain to a neighborhood of one of the corners of the equilateral . Since the angle of the corner of equals which is also the angle of the equilateral, we can use Schwarz reflection principle to extend to a conformal map in a neighborhood of , as shown in the figure above. We first reflect w.r.t. two each of the edges touching the corner, and then reflect w.r.t the new line boundary (similarly to other, noncorner, boundary charts). The extended map is analytic (and conformal) also at the corner point because it is continuous there and conformal in its neighborhood.
To finish the proof we need to show that are homeomorphic mappings from to . Note that is an affine map that is composed of two affine maps. The first, denoted by , takes in to the coordinate chart’s triangle , and the second, , maps to . The first affine map is orientation preserving and nondegenerate (and with bounded conformal distortion) for sufficiently large due to Lemma 2.5. The second affine map was proven above (by showing that the determinant of its Jacobian is positive) to be orientation preserving and nondegenerate as well for all faces and for sufficiently large . Therefore, is orientation preserving nondegenerate as a composition of two orientationpreserving and nondegenerate affine maps. Since also maps the boundary of bijectively onto the boundary of (by definition of ), it is a global bijection (see [17] for a detailed proof). Since it is simplicial, its inverse is also continuous, a fact which makes it a homeomorphism. ∎
A comments is in order. One can build the charts based on an already subdivided version of the surface ,
, and achieve better estimates of the conformal distortion of the maps
. For example, if one takes the charts defined by , the vertices are all flat, i.e. , and therefore triangles associated with the corresponding charts will have conformal distortion of order .4. Approximation
Our next step is showing that if we are able to put our hands on a series of simplicial maps with discrete conformal distortion that goes uniformly to one, then this series has to uniformly converge to the uniformization map . Note that this result applies to any such series, not only the ones built in Theorem 3.1.
Theorem 4.1.
Let be the uniformization map of the polyhedral surface . Let , for some series , where is some constant.
Then, converges uniformly in any compact subset of to the identity map .
Proof.
Define .
We start by showing that is quasiconformal with some global distortion bound .
First, since is an orientationpreserving homeomorphism, is orientationpreserving homeomorphism. By assumption is an orientationpreserving homeomorphism and so is an orientationpreserving homeomorphism. Let us observe that can be written as
where is the chart associated with the face . The part that is marked with is a conformal map over (by the definition of the uniformization map). Next, recall that consists of a composition of a rigid maps and power map , which is conformal except at the origin. Therefore, is conformal over the domain . Lastly, the part is defined over the set and can be written as a composition of two affine maps (as before) the first, denoted by , takes in to the coordinate chart triangle ,where , are the (ordered) vertices of face . The second affine map, , maps to . is quasiconformal, for some independent of by Lemma 2.5. is quasiconformal, again independently of , due to the assumption that , . Then, is quasiconformal, for all faces and all . Denote the domain
The set , which consists of the image under of the edges of is a union of analytic arcs. By extension theorem for quasiconformal mappings (see e.g., Theorem 8.2, page 42, and Theorem 8.3, page 45, in [15]) the conformal distortion of over the whole is again. Since bounded quasiconformal maps form a normal family there exists a subsequence that converges locally uniformly to a quasiconformal map or a constant (see Theorem 3.1.3, page 49 in [6]). Denote the limit map by . Since fixes the corners of , is quasiconformal. Next we show that is a conformal map. It is enough to prove that is 1quasiconformal (see e.g., Theorem 5.1, page 28 in [15]). For that end, take an arbitrary domain such that . Now, we look at . Similar to above, it consists of a composition of conformal maps ( and the inverse of the power map ), and an affine map that can be represented as composition of two affine maps and . The conformal distortion of equals by assumption. The affine map is defined by sampling the power map , at the corners of (the origin is placed at vertex ) and extending linearly. Since, by the intersection with the domain , we sample at some bounded (from below) distance from the origin (vertex ), Lemma 3.2 indicates that . This implies that the conformal distortion of
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