# Boundary First Flattening

arXiv: https://arxiv.org/abs/1704.06873 · HTML: https://mlsanity.com/paper/1704.06873/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Apr 2017.

## Built on by (most cited first, 17 of 17)

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- [MeshTailor: Cutting Seams via Generative Mesh Traversal](https://mlsanity.com/paper/2603.27309/) (2026)
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- [Initialization and Alignment for Adversarial Texture Optimization](https://mlsanity.com/paper/2207.14289/) (2022)
- [From Extrinsic to Intrinsic: Geodesic-Guided Representation Learning for 3D Geometric Data](https://mlsanity.com/paper/2606.02268/) (2026)
- [Adaptive area-preserving parameterization of open and closed anatomical surfaces](https://mlsanity.com/paper/2111.04265/) (2021)
- [Generating 3D Bio-Printable Patches Using Wound Segmentation and Reconstruction to Treat Diabetic Foot Ulcers](https://mlsanity.com/paper/2203.03814/) (2022)
- [Advances in Neural 3D Mesh Texturing: A Survey](https://mlsanity.com/paper/2606.00137/) (2026)
- [Monte Carlo Steklov Operators for Large-Scale Geometry Processing in the Wild](https://mlsanity.com/paper/2606.05581/) (2026)
- [Continuous Neural Reparameterization as a Deep Geometric Prior for Robust Fixed-Chart UV Repair](https://mlsanity.com/paper/2606.10050/) (2026)
- [SeamGen: Artist-Aligned UV Seam Generation via Graph Flow Matching](https://mlsanity.com/paper/2607.12379/) (2026)

## Abstract (arXiv)

A conformal flattening maps a curved surface to the plane without distorting angles---such maps have become a fundamental building block for problems in geometry processing, numerical simulation, and computational design. Yet existing methods provide little direct control over the shape of the flattened domain, or else demand expensive nonlinear optimization. Boundary first flattening (BFF) is a linear method for conformal parameterization which is faster than traditional linear methods, yet provides control and quality comparable to sophisticated nonlinear schemes. The key insight is that the boundary data for many conformal mapping problems can be efficiently constructed via the Cherrier formula together with a pair of Poincare-Steklov operators; once the boundary is known, the map can be easily extended over the rest of the domain. Since computation demands only a single factorization of the real Laplace matrix, the amortized cost is about 50x less than any previously published technique for boundary-controlled conformal flattening. As a result, BFF opens the door to real-time editing or fast optimization of high-resolution maps, with direct control over boundary length or angle. We show how this method can be used to construct maps with sharp corners, cone singularities, minimal area distortion, and uniformization over the unit disk; we also demonstrate for the first time how a surface can be conformally flattened directly onto any given target shape.