A Nonlinear Extension of the Variable Projection (VarPro) Method for NURBS-based Conformal Surface Flattening

Abstract

In the field of computer graphics, conformal surface flattening has been widely studied for tasks such as texture mapping, geometry processing, and mesh generation. Typically, existing methods aim to flatten a given input geometry while preserving conformality as much as possible, meaning the result is only as conformal as possible. By contrast, this study focuses on surfaces that can be flattened conformally without singularities, making the process a coupled problem: the input (or target) surface must be recursively refined while its flattening is computed. Although the uniformization theorem or the Riemann mapping theorem guarantees the existence of a conformal flattening for any simply connected, orientable surface, those theorems permit singularities in the flattening. If singularities are not allowed, only a special class of surfaces can be conformally flattened-though many practical surfaces do fall into this class. To address this, we develop a NURBS-based approach in which both the input surface and its flattening are refined in tandem, ensuring mutual conformality. Because NURBS surfaces cannot represent singularities, the resulting pair of surfaces is naturally singularity-free. Our work is inspired by the form-finding method by [Miki and Mitchell 2022, 2024], which solves bilinear PDEs by iteratively refining two surfaces together. Building on their demonstration of the effectiveness of variable projection (VarPro), we adopt a similar strategy: VarPro alternates between a linear projection and a nonlinear iteration, leveraging a partially linear (separable) problem structure. However, since our conformal condition separates into two nonlinear subproblems, we introduce a nonlinear extension of VarPro. Although this significantly increases computational cost, the quality of the results is noteworthy.