# Boundary First Flattening

## TL;DR

Boundary First Flattening (BFF) is a fast, linear conformal mapping method that offers direct boundary control and high-quality results, enabling real-time editing and complex shape conformal flattening.

## Contribution

BFF introduces a linear, boundary-controlled conformal flattening technique that is significantly faster than existing methods while maintaining high quality and flexibility.

## Key findings

- BFF is about 50 times faster than previous boundary-controlled methods.
- It enables real-time editing and optimization of high-resolution conformal maps.
- The method can produce maps with sharp corners, cone singularities, and conformal flattening onto arbitrary shapes.

## Abstract

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.

---
Source: https://tomesphere.com/paper/1704.06873