Blended Chart Surfaces

Abstract

A surface representation suitable for geometry processing should be compact and explicit, provide global smoothness guarantees, support a wide range of surface topologies, and offer reliable access to differential quantities such as normals and surface energies, while remaining compatible with modern differentiable optimization. Yet existing neural representations typically sacrifice one or more of these properties: implicit fields typically require iso-surfacing for downstream use, while explicit neural maps are constrained by canonical-domain parametrizations and/or exhibit seam artifacts between local charts. We introduce Blended Chart Surfaces (BCS), a compact, network-free, explicit representation that is smooth by construction and anchored to user-provided topology. Given a coarse proxy mesh encoding the intended surface topology and approximate geometry, Blended Chart Surfaces jointly optimize for a polynomial map at each proxy vertex using an off-the-shelf optimizer to fit to an implicit target shape, avoiding the need for an input parametrization. Neighboring maps are fused using a smooth ‘one-ring coordinate’ blending scheme, decoupling surface topology and coarse geometry (carried by the proxy) from geometric details (carried by the smooth local patches). The resulting surface is globally smooth, fully differentiable, and enables stable evaluation of positions and derivatives, making differential quantities and surface energies directly accessible. Additionally, our construction is equivariant to rigid motions and scaling of the proxy mesh. We evaluate Blended Chart Surfaces on surfaces spanning varying topology and geometric complexity, and compare against explicit alternatives including interpolating-function baselines and mesh-displacement MLPs. Across these, Blended Chart Surfaces achieves a favorable trade-off among compactness, simplicity, access to differential quantities, and expressivity while remaining smooth across patch boundaries. Code will be released.

Contributions

1. We introduce a compact, explicit surface representation as a set of overlapping local polynomial patches blended with partition-of-unity weights, yielding global smoothness and enabling end-to-end differentiation and surface fitting.
2. We encode a target surface on a given coarse mesh by optimizing patch coefficients directly, without expecting any global charting as input, while supporting a broad range of object topologies specified by the proxy — naturally accommodating difficult topologies such as open boundaries and non-orientable surfaces.
3. We admit progressive levels of detail by increasing polynomial degree (e.g., constant, linear, quadratic, cubic) and/or by refining the proxy, providing a continuum between compactness and faithful reconstructions (i.e., surface fitting).

Method Overview

2D Results — Blended Chart Curves

The construction first specializes to the 1D/2D curve case: a vertex function is assigned to each vertex of a coarse proxy polygon, and neighboring functions are fused with a smooth blending scheme to produce a curve that is C^∞ by construction and equivariant to rigid motions and scaling of the proxy.

3D Results — Blended Chart Surfaces

In 3D, a C^∞-continuous local map is associated with each vertex of a coarse triangle mesh using one-ring coordinates, and neighboring maps are fused with a partition-of-unity blending function. Fitting optimizes the per-vertex polynomial coefficients directly against an implicit target, with no global parametrization required.

Quantitative Evaluation

Acknowledgements

BibTeX

@article{williamson2026blended,
  title   = {Blended Chart Surfaces: A Seamless Explicit
             Representation for Smooth Surface Fitting},
  author  = {Williamson, Romy and Mitra, Niloy J.},
  journal = {Computer Graphics Forum},
  year    = {2026},
  note    = {Proceedings of Pacific Graphics 2026},
}