Blended Chart Surfaces: Smooth & Differentiable

• Blended Chart Surfaces are explicit representations combining a triangulated proxy mesh with local polynomial maps to capture detailed geometry.
• They yield globally smooth, fully differentiable surfaces by blending per-vertex charts using a C∞ partition of unity.
• This method achieves compact parameter storage and equivariance under rigid motions and scaling, enhancing efficient geometry processing.

Blended Chart Surfaces are a compact, network-free, explicit surface representation in which a triangulated proxy mesh supplies topology and approximate geometry, while a low-degree polynomial map attached to each proxy vertex carries local geometric detail. Neighboring maps are fused by a smooth one-ring coordinate blending scheme, producing a globally smooth, fully differentiable surface with direct access to normals, curvature, and surface energies, and the construction is equivariant to rigid motions and uniform scaling of the proxy mesh (Williamson et al., 16 Jun 2026).

1. Problem setting and relation to earlier chart-based surface models

A surface representation for geometry processing is described as needing to be compact and explicit, to provide global smoothness guarantees, to support a wide range of surface topologies, to offer reliable access to differential quantities such as normals and surface energies, and to remain compatible with modern differentiable optimization. In the same formulation, implicit fields are characterized as typically requiring iso-surfacing for downstream use, whereas explicit neural maps are characterized as either constrained by canonical-domain parametrizations or prone to seam artifacts between local charts (Williamson et al., 16 Jun 2026).

Blended Chart Surfaces arise within a broader line of multi-chart and patch-blending methods, but they address a different operating point. Earlier multi-chart generative modeling represented genus-zero shapes by multiple conformal charts, sampled each chart on a uniform grid, concatenated the resulting chart tensors, and used chart normalization together with scale-translation rigidity to guarantee unique reconstruction after learning (Ben-Hamu et al., 2018). In contrast, Blended Chart Surfaces avoid an input parametrization and are anchored directly to a user-provided proxy mesh of arbitrary topology (Williamson et al., 16 Jun 2026).

A second neighboring line of work uses blended local interpolants over polyhedral control meshes. “Polyhedral design with blended $n$-sided interpolants” constructs smoothly connecting quadrilateral patches by blending local, multi-sided quadratic interpolants, including special parameterization for the non-four-sided case through rational curves (Salvi, 27 Jan 2026). Another distinct neighboring formulation, the Method of Matched Sections, reconstructs energetically optimal fair surfaces by reducing a thin-plate problem to coupled one-dimensional beam-like problems and globally enforcing continuity of displacement, curvature, and shear-consistent quantities across element boundaries (Orynyak et al., 6 Dec 2025). These precedents clarify that “blended charts” can denote several different constructions; in Blended Chart Surfaces, the defining mechanism is a per-vertex polynomial patch blended over the proxy one-rings by a $C^\infty$ partition of unity.

2. Local polynomial maps on proxy one-rings

Let the proxy mesh be $M=(V,E,F)$, a triangulated surface with arbitrary topology. At each proxy vertex $v_i$, with one-ring of valence $\eta_i$, the one-ring is first flattened to an equilateral triangle fan in $\mathbb R^2$ by a cone-flattening. In these local coordinates $(u,v)\in\Omega\subset\mathbb R^2$, the local patch is a degree-$d$ polynomial

$m_i(u,v)=A_i\,\Phi_d(u,v)^T\in\mathbb R^3,$

with monomial basis

$$\Phi_d(u,v)=\bigl[1,\;u,\;v,\;u^2,\;uv,\;v^2,\;\dots,\;u^d,\;u^{d-1}v,\;\dots,\;v^d\bigr].$$

The unknown coefficient matrix is $C^\infty$0 (Williamson et al., 16 Jun 2026).

Equivariance is built into the embedding of this local polynomial back into $C^\infty$1. Each local patch is expressed as

$C^\infty$2

where $C^\infty$3 is a local frame aligned to the vertex normal at $C^\infty$4, and

$C^\infty$5

defines a local scale proportional to the adjacent proxy edge lengths (Williamson et al., 16 Jun 2026).

This construction separates roles that are often entangled in other explicit surface models. The proxy carries topology and coarse geometry, while the optimized polynomial coefficients carry local geometric detail. Because the representation is explicit, evaluating the surface does not require extracting an isosurface, and because the per-vertex maps are low-degree polynomials rather than learned MLPs, the parameterization remains compact in the literal sense of stored coefficients.

3. One-ring coordinates and $C^\infty$6 blending

If $C^\infty$7 lies in a proxy triangle $C^\infty$8, then $C^\infty$9 lies in the one-rings of exactly the three vertices $M=(V,E,F)$0. For each such center $M=(V,E,F)$1, one computes a one-ring coordinate by first mapping the physical triangle affinely to a unit-side-length equilateral triangle in $M=(V,E,F)$2, then expressing $M=(V,E,F)$3 in polar form $M=(V,E,F)$4 about the central vertex, and finally rescaling the angular coordinate by $M=(V,E,F)$5. The resulting chart coordinate is

$M=(V,E,F)$6

These flattenings are diffeomorphisms away from the vertex itself and agree smoothly on overlaps, so they serve as local charts of a $M=(V,E,F)$7 atlas (Williamson et al., 16 Jun 2026).

The surface over $M=(V,E,F)$8 is then defined by blending the three local patches: $M=(V,E,F)$9 with partition-of-unity weights

$v_i$0

The radial bump is constructed from

$v_i$1

with overlap parameter $v_i$2 (Williamson et al., 16 Jun 2026).

The smoothness claim follows directly from the ingredients. Each triangle-wise map $v_i$3 is a finite linear combination of compositions of $v_i$4 functions: the one-ring flattenings, the polynomials $v_i$5, the rigid-motion insertions, and the bump functions. Because the transition maps between overlapping one-rings are $v_i$6 diffeomorphisms and the weights form a partition of unity, the resulting surface is $v_i$7 across triangle boundaries (Williamson et al., 16 Jun 2026). In this framework, seams are not removed by post hoc stitching; they are absent by construction.

4. Fitting to implicit targets

Blended Chart Surfaces are optimized against an implicit target, typically given as a signed-distance field $v_i$8. Each proxy triangle is sampled at quadrature points $v_i$9 in the equilateral domain $\eta_i$0 with weights $\eta_i$1. The primary fitting objective is the average unsigned signed-distance magnitude,

$\eta_i$2

To discourage fold-overs and align normals, the construction adds

$\eta_i$3

where the exact unit normal is

$\eta_i$4

A fuller objective may include distortion terms,

$\eta_i$5

and the optimization is carried out by standard Adam with back-propagation through all polynomials and blend weights (Williamson et al., 16 Jun 2026).

Two structural consequences are notable. First, the representation is explicit even though it is fitted to an implicit target: the target appears only in the loss, not in the output form. Second, the absence of an input parametrization is substantive rather than rhetorical. The fitting pipeline starts from the proxy and the implicit field, rather than from a canonical two-dimensional domain or a precomputed surface atlas.

5. Differential quantities, equivariance, and empirical behavior

A principal motivation for the representation is reliable access to differential quantities. Because the surface map is explicit and smooth, derivatives can be obtained directly by automatic differentiation. The first and second fundamental forms are

$\eta_i$6

from which mean and Gauss curvature follow as

$\eta_i$7

The paper characterizes these as smooth fields on the surface (Williamson et al., 16 Jun 2026).

Equivariance is built into the representation. Under a global rigid motion $\eta_i$8, the local frames transform as $\eta_i$9, the local scales remain unchanged, and each patch satisfies

$\mathbb R^2$0

Because the blend weights depend only on the equilateral parameter, the blended triangle map transforms as $\mathbb R^2$1. Under a uniform scaling $\mathbb R^2$2, the local scales satisfy $\mathbb R^2$3, hence $\mathbb R^2$4 and $\mathbb R^2$5 (Williamson et al., 16 Jun 2026).

The reported empirical profile emphasizes compactness and fidelity. Blended Chart Surfaces store only $\mathbb R^2$6 scalar coefficients; for quadratic patches ($\mathbb R^2$7) this is $\mathbb R^2$8. For a genus-4 “Fertility” model with $\mathbb R^2$9, this yields $(u,v)\in\Omega\subset\mathbb R^2$0 numbers, whereas a small displacement-MLP on the same proxy uses $(u,v)\in\Omega\subset\mathbb R^2$1 parameters. With 250 vertices and quadratic patches, the reported average unsigned-distance errors are $(u,v)\in\Omega\subset\mathbb R^2$2 after unit-box normalization, and these results are reported as outperforming both quadratic vertex-centric interpolating splines and displacement-MLPs (Williamson et al., 16 Jun 2026). The same evaluation also contrasts continuity classes: by design the representation is $(u,v)\in\Omega\subset\mathbb R^2$3, whereas barycentric partitions are described as only $(u,v)\in\Omega\subset\mathbb R^2$4, simple radial weights as leaving gaps, and a trigonometric bump as only $(u,v)\in\Omega\subset\mathbb R^2$5.

6. Interpretation, adjacent meanings of “chart blending,” and limitations

The term “chart” is overloaded across several literatures, and this has practical consequences for interpretation. In Blended Chart Surfaces, a chart is a local geometric parameterization centered at a proxy vertex and blended over overlapping one-rings. In multi-chart generative modeling, the charts are conformal toric parameterizations of genus-zero surfaces arranged into an image-like tensor for GAN training, with chart blending performed afterward by weighted averaging on a fixed template mesh (Ben-Hamu et al., 2018). In polyhedral design with blended $(u,v)\in\Omega\subset\mathbb R^2$6-sided interpolants, local quadratic or multi-sided interpolants are blended over quadrilateral or kite-shaped subdomains to interpolate a given closed mesh (Salvi, 27 Jan 2026). In thin-plate reconstruction via the Method of Matched Sections, “blending” refers to assembling energetically optimal fair surfaces from matched element sections under a biharmonic variational principle (Orynyak et al., 6 Dec 2025). In visualization, by contrast, overlapped charts denote transparent graphical layers, and blending refers to alpha compositing and color-name aware optimization for perceptual discriminability rather than geometric surface construction (Lu et al., 2024).

Several limitations also follow from the stated formulation. Because the method is anchored to user-provided topology and begins from a coarse proxy mesh encoding the intended surface topology and approximate geometry, topology is not inferred by the representation itself (Williamson et al., 16 Jun 2026). A plausible implication is that proxy-topology errors persist unless the proxy is changed. Likewise, the explicit inclusion of $(u,v)\in\Omega\subset\mathbb R^2$7 “to discourage fold-overs and align normals” suggests that global smoothness of the blend does not by itself guarantee a satisfactory fit during optimization; the fitting objective must still control local geometric behavior (Williamson et al., 16 Jun 2026).

Within those constraints, the representation occupies a specific niche. It is explicit rather than implicit, smooth by construction rather than stitched after fitting, compact in stored parameters, and directly compatible with differentiable optimization. The central claim is therefore not merely that local patches are blended, but that smoothness, differential access, and equivariance are all encoded in the representation itself rather than added as downstream corrections.