Barycentric Discrete Conformal Maps

• Barycentric discrete conformal maps are defined by combining piecewise-linear discrete conformal mappings with Riemannian barycentric coordinates, linking simplicial geometry to smooth structures.
• They employ vertex-based scale factors and affine finite-element techniques to achieve convergence and rigidity, thereby approximating classical conformal maps and establishing Möbius equivalence in higher dimensions.
• The approach integrates linear variational formulations and convex energy minimization, enabling efficient computation in applications like Riemann map approximations and circle packing convergence.

Barycentric discrete conformal maps are discrete conformal mappings in which simplicial, piecewise-flat geometry is related to smooth geometry through barycentric constructions. In the most explicit formulation, they are maps of the form

$\Phi_n=\bar\Psi_n\circ \phi_n\circ \Psi_n^{-1},$

where $\phi_n$ is a piecewise linear discrete conformal map between triangulated PL surfaces and $\Psi_n,\bar\Psi_n$ are Riemannian barycentric maps between simplices and the underlying manifolds (Glickenstein et al., 22 Jul 2025). The term also has a broader, adjacent usage in work where the map is represented by vertex values and affine interpolation on triangles, or where discrete conformality is encoded by vertex-based scale factors on edges rather than by classical barycentric averaging formulas. In that broader sense, the subject links linear finite-element approximations of Riemann maps (Dym et al., 2017), vertex-scaling PL conformal maps on triangular lattices (Bücking, 2015), and higher-dimensional rigidity theorems for vertex-scaled simplicial complexes (Pinkall et al., 2019).

1. Terminology, scope, and basic algebraic forms

A central modern formulation starts with a triangulated manifold $(M,T)$ and discrete conformal parameters

$$\alpha \in \mathbb{R}^{|V|}, \qquad \eta \in \mathbb{R}^{|E|}.$$

A discrete conformal factor $f \in \mathbb{R}^{|V|}$ determines edge lengths by

$$\ell_{ij}^2 \;=\; \alpha_i e^{2f_i} + \alpha_j e^{2f_j} + 2\eta_{ij} e^{f_i+f_j},$$

for every edge $e_{ij}$, with admissible factors restricted to the domain for which all simplices are nondegenerate. This unified form includes circle packing, vertex scaling, and related Euclidean discrete conformal structures (Glickenstein et al., 22 Jul 2025).

A complementary two-dimensional definition describes a discrete conformal PL-mapping $g:T_S\to\mathbb C$ on a triangular mesh as a continuous, orientation-preserving, locally homeomorphic, piecewise-linear map for which there exists a vertex function $u:V_S\to\mathbb R$ such that for every edge $\phi_n$0,

$\phi_n$1

Equivalently,

$\phi_n$2

Here the conformal data live on vertices, while the map itself is assembled triangle-by-triangle as a PL map (Bücking, 2015).

The literature does not use the term “barycentric discrete conformal map” uniformly. Some works define it explicitly through Riemannian barycentric coordinates on manifolds, while others are connected to it only through affine interpolation on simplices or vertex-centered scaling laws. This suggests a family resemblance rather than a single universal definition.

2. Linear variational principles and affine finite-element constructions

One important precursor is the linear variational characterization of the Riemann map from a bounded Lipschitz domain $\phi_n$3 in the plane to a triangle $\phi_n$4. The classical Dirichlet problem is relaxed by replacing the nonlinear boundary homeomorphism constraint with the linear condition that each boundary arc $\phi_n$5 maps into the supporting line $\phi_n$6 of a target edge:

$\phi_n$7

Over the resulting relaxed admissible set, the Dirichlet energy

$\phi_n$8

has the Riemann map $\phi_n$9 as its unique minimizer. The paper proves that this relaxation is tight: the relaxed Plateau problem still selects the Riemann mapping (Dym et al., 2017).

The discrete formulation uses a triangulation $\Psi_n,\bar\Psi_n$0 and the FEM space $\Psi_n,\bar\Psi_n$1 of continuous piecewise-linear functions. A discrete map is written in nodal form,

$\Psi_n,\bar\Psi_n$2

and the boundary conditions become affine constraints on boundary vertices,

$\Psi_n,\bar\Psi_n$3

The discrete Dirichlet energy is the FEM quadratic form

$\Psi_n,\bar\Psi_n$4

The resulting optimization problem is a strictly convex quadratic program with linear constraints, solved by a sparse linear system. In this construction, the map is determined by vertex values and linearly interpolated over triangles, so the connection to barycentric discrete conformal mapping is affine and finite-element based rather than an explicit use of barycentric coordinates in the mesh-parameterization sense. In the Euclidean orbifold case, when $\Psi_n,\bar\Psi_n$5 is an equilateral triangle or a right-angled isosceles triangle, the discrete problem is exactly the Orbifold-Tutte algorithm; for Delaunay triangulations this map is known to be bijective. The convergence theory is correspondingly strong: $\Psi_n,\bar\Psi_n$6 convergence holds for regular triangulations even when they are non-Delaunay, while uniform convergence is available in the 3-connected Delaunay orbifold setting (Dym et al., 2017).

3. Barycentric maps on Riemannian surfaces

The explicit modern theory constructs barycentric discrete conformal maps on triangulated Riemannian surfaces by combining PL discrete conformal maps with Riemannian barycentric coordinates. For points $\Psi_n,\bar\Psi_n$7 in a complete Riemannian manifold $\Psi_n,\bar\Psi_n$8 and barycentric weights $\Psi_n,\bar\Psi_n$9, the associated energy is

$(M,T)$0

When the vertices lie in a sufficiently small convex ball, this energy has a unique minimizer, and the Riemannian barycentric map $(M,T)$1 sends $(M,T)$2 to that minimizer. Its image is the Karcher simplex. Given barycentric maps $(M,T)$3 and $(M,T)$4, together with a PL discrete conformal map $(M,T)$5, the barycentric discrete conformal map is

$(M,T)$6

On each closed simplex, $(M,T)$7 is a diffeomorphism (Glickenstein et al., 22 Jul 2025).

The nondegeneracy condition is $(M,T)$8-fullness. A simplex $(M,T)$9 is $$\alpha \in \mathbb{R}^{|V|}, \qquad \eta \in \mathbb{R}^{|E|}.$$0-full if all edges satisfy $$\alpha \in \mathbb{R}^{|V|}, \qquad \eta \in \mathbb{R}^{|E|}.$$1 and

$$\alpha \in \mathbb{R}^{|V|}, \qquad \eta \in \mathbb{R}^{|E|}.$$2

Fullness prevents skinny or degenerate simplices, gives a positive lower bound on simplex heights, and yields metric distortion estimates for barycentric maps. The key rigidity input is Local Discrete Conformal Rigidity (LDCR), an abstract replacement for hexagonal rigidity in circle packing. In its equivalent form, if

$$\alpha \in \mathbb{R}^{|V|}, \qquad \eta \in \mathbb{R}^{|E|}.$$3

then vertices $$\alpha \in \mathbb{R}^{|V|}, \qquad \eta \in \mathbb{R}^{|E|}.$$4 in sufficiently large combinatorial neighborhoods satisfy

$$\alpha \in \mathbb{R}^{|V|}, \qquad \eta \in \mathbb{R}^{|E|}.$$5

and likewise

$$\alpha \in \mathbb{R}^{|V|}, \qquad \eta \in \mathbb{R}^{|E|}.$$6

The central metric estimate states that the pullback metric of $$\alpha \in \mathbb{R}^{|V|}, \qquad \eta \in \mathbb{R}^{|E|}.$$7 is approximately scalar:

$$\alpha \in \mathbb{R}^{|V|}, \qquad \eta \in \mathbb{R}^{|E|}.$$8

From this, one obtains two levels of convergence. For an admissible sequence, the maps $$\alpha \in \mathbb{R}^{|V|}, \qquad \eta \in \mathbb{R}^{|E|}.$$9 have a subsequence converging uniformly on compact subsets. For a proper admissible sequence, there exists a positive continuous function $f \in \mathbb{R}^{|V|}$0 such that

$f \in \mathbb{R}^{|V|}$1

in $f \in \mathbb{R}^{|V|}$2 on compact subsets, hence the limit is conformal. The theory is designed to cover surfaces with or without boundary and a variety of discrete conformal structures within one framework (Glickenstein et al., 22 Jul 2025).

4. Vertex scaling, convex energies, and approximation of smooth conformal maps

A second major strand studies discrete conformal maps through conformally equivalent triangular meshes. On a subcomplex $f \in \mathbb{R}^{|V|}$3 of a regular triangular lattice, a vertex function $f \in \mathbb{R}^{|V|}$4 determines rescaled edge lengths

$f \in \mathbb{R}^{|V|}$5

Such a field comes from a discrete conformal PL-map if and only if two geometric conditions hold: the rescaled edge lengths satisfy the triangle inequalities on every triangle, and the angles in the incident triangles around each interior vertex sum to $f \in \mathbb{R}^{|V|}$6. The latter is written using a specific angle function $f \in \mathbb{R}^{|V|}$7 derived from the half-angle formula and the logarithmic edge-ratio quantities

$f \in \mathbb{R}^{|V|}$8

This is a mesh-based characterization of discrete conformality rather than a barycentric-coordinate formula (Bücking, 2015).

The variational framework uses a convex functional $f \in \mathbb{R}^{|V|}$9 whose Euler–Lagrange equations are exactly the angle-sum equations. For prescribed boundary values of the scale factors, the unknown interior values are obtained as the unique minimizer of $$\ell_{ij}^2 \;=\; \alpha_i e^{2f_i} + \alpha_j e^{2f_j} + 2\eta_{ij} e^{f_i+f_j},$$0. When a smooth conformal map $$\ell_{ij}^2 \;=\; \alpha_i e^{2f_i} + \alpha_j e^{2f_j} + 2\eta_{ij} e^{f_i+f_j},$$1 is given, the discrete boundary data are prescribed by

$$\ell_{ij}^2 \;=\; \alpha_i e^{2f_i} + \alpha_j e^{2f_j} + 2\eta_{ij} e^{f_i+f_j},$$2

After fixing a basepoint and an edge direction for normalization, the resulting discrete conformal PL-map $$\ell_{ij}^2 \;=\; \alpha_i e^{2f_i} + \alpha_j e^{2f_j} + 2\eta_{ij} e^{f_i+f_j},$$3 approximates $$\ell_{ij}^2 \;=\; \alpha_i e^{2f_i} + \alpha_j e^{2f_j} + 2\eta_{ij} e^{f_i+f_j},$$4 on increasingly fine subcomplexes of a strictly acute triangular lattice (Bücking, 2015).

The approximation theorem is quantitative. For all sufficiently small $$\ell_{ij}^2 \;=\; \alpha_i e^{2f_i} + \alpha_j e^{2f_j} + 2\eta_{ij} e^{f_i+f_j},$$5, there exists a unique discrete conformal PL-map $$\ell_{ij}^2 \;=\; \alpha_i e^{2f_i} + \alpha_j e^{2f_j} + 2\eta_{ij} e^{f_i+f_j},$$6 with the prescribed boundary scale factors, and the estimates are

$$\ell_{ij}^2 \;=\; \alpha_i e^{2f_i} + \alpha_j e^{2f_j} + 2\eta_{ij} e^{f_i+f_j},$$7

$$\ell_{ij}^2 \;=\; \alpha_i e^{2f_i} + \alpha_j e^{2f_j} + 2\eta_{ij} e^{f_i+f_j},$$8

and, on triangle interiors,

$$\ell_{ij}^2 \;=\; \alpha_i e^{2f_i} + \alpha_j e^{2f_j} + 2\eta_{ij} e^{f_i+f_j},$$9

Thus the PL maps converge uniformly in $e_{ij}$0 with error of order $e_{ij}$1, while the scale factors converge with order $e_{ij}$2. The strict acuteness assumption is essential because it underlies a monotonicity lemma for the angle-sum operator and the barrier-function argument used in the proof (Bücking, 2015).

5. Rigidity, Möbius equivalence, and the higher-dimensional boundary of the theory

In dimensions $e_{ij}$3, the corresponding vertex-scaling theory becomes rigid. Two combinatorially equivalent simplicial complexes in $e_{ij}$4 are discretely conformally equivalent if corresponding edge lengths satisfy

$e_{ij}$5

This is the same geometric-mean scaling law seen in two dimensions, but its global consequences are different (Pinkall et al., 2019).

The main theorem states that if $e_{ij}$6, then two locally Delaunay discrete domains in $e_{ij}$7 are discretely conformally equivalent if and only if they are Möbius equivalent. The proof proceeds in two stages. First, a simplex-level lemma shows that two $e_{ij}$8-simplices satisfy the vertex-scaling law if and only if there exists a Möbius transformation taking one simplex to the other vertexwise. Second, the local Delaunay condition implies that after inversion at an interior vertex, the star of that vertex becomes a convex polyhedron. The facets of the inverted stars are pairwise similar, so Cauchy rigidity forces a single similarity, and undoing the inversion yields a single Möbius map on the entire star. Connectivity of the interior 1-skeleton then propagates this map globally (Pinkall et al., 2019).

This rigidity result is the discrete analogue of smooth Liouville rigidity. It corrects a common overextension of two-dimensional intuition: vertex-scaled simplicial conformality is flexible enough to approximate nontrivial conformal maps on surfaces, but in dimensions greater than two it collapses, under the local Delaunay and domain hypotheses, to Möbius geometry. The same paper also gives a hyperbolic interpretation: two triangulated piecewise Euclidean manifolds are discretely conformally equivalent if and only if they are isometric with respect to the induced hyperbolic metrics (Pinkall et al., 2019).

6. Applications, special cases, and adjacent computational frameworks

A direct application of the linear variational triangle-mapping theory is approximation of the Riemann map between two bounded Lipschitz domains by composition through a common triangle. If $e_{ij}$9 and $g:T_S\to\mathbb C$0 are the discrete conformal maps, then

$g:T_S\to\mathbb C$1

converges uniformly to the Riemann map $g:T_S\to\mathbb C$2. The reduction is explicit: map each domain to the triangle, then compose via the inverse triangle map. Because the method reduces to a sparse linear solve, the paper highlights that even a six-million-vertex Koch snowflake triangulation can be handled in about two minutes with Matlab’s linear solver (Dym et al., 2017).

The 2025 barycentric theory recovers the Rodin–Sullivan circle packing theorem as a special case. For a bounded simply connected domain $g:T_S\to\mathbb C$3, hexagonal circle packings with radius $g:T_S\to\mathbb C$4 give a generalized triangulated exhaustion of $g:T_S\to\mathbb C$5, a proper admissible sequence for $g:T_S\to\mathbb C$6, and hence a subsequence of circle packing maps converging uniformly on compacta to a Riemann map $g:T_S\to\mathbb C$7. The same framework is presented as conceptually subsuming vertex-scaling convergence results associated with Bücking, Gu–Luo–Wu, Luo–Sun–Wu, Luo–Wu–Zhu, and Wu–Zhu (Glickenstein et al., 22 Jul 2025).

There are also adjacent computational models that are barycentric in spirit but not instances of the triangle-mesh formalism. A meshless method for point clouds approximates a surface by an $g:T_S\to\mathbb C$8-neighborhood sampled with a cubic lattice, computes discrete harmonic maps on the resulting graph, and obtains conformal maps by minimizing Dirichlet energy while deforming the target surface of constant curvature. The method uses a graph Laplacian on the cubic grid, target-moduli optimization for spheres, rectangles, flat tori, and hyperbolic surfaces, and trilinear interpolation back to the point cloud. The paper explicitly notes that it does not use the barycentric discrete conformal map formalism directly, although its vertex-balance conditions and variational characterization of harmonicity are barycentric in spirit (Wu et al., 2020).

The resulting landscape is therefore heterogeneous but structurally coherent. Some theories are explicitly barycentric, in the sense of Riemannian barycentric coordinates composed with PL discrete conformal maps. Others are affine/FEM or vertex-scaling theories that become “barycentric” only at the level of simplex interpolation or vertex-centered metric data. The common technical themes are local simplicial geometry, vertex-based conformal parameters, variational control through Dirichlet or convex energies, and convergence or rigidity theorems that specify when the discrete model approximates classical conformal mapping and when it reduces to Möbius geometry.