# A Linear Formulation for Disk Conformal Parameterization of Simply-Connected Open Surfaces

arXiv: https://arxiv.org/abs/1508.00396 · HTML: https://mlsanity.com/paper/1508.00396/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Aug 2015.

## Built on by (most cited first, 5 of 5)

- [A unifying framework for n-dimensional quasi-conformal mappings](https://mlsanity.com/paper/2110.10437/) (2021)
- [Automatic Landmark Detection and Registration of Brain Cortical Surfaces via Quasi-Conformal Geometry and Convolutional Neural Networks](https://mlsanity.com/paper/2208.07010/) (2022)
- [Adaptive area-preserving parameterization of open and closed anatomical surfaces](https://mlsanity.com/paper/2111.04265/) (2021)
- [Gaussian Differentially Private Human Faces Under a Face Radial Curve Representation](https://mlsanity.com/paper/2409.08301/) (2024)
- [Variational Geometry-aware Neural Network based Method for Solving High-dimensional Diffeomorphic Mapping Problems](https://mlsanity.com/paper/2511.01911/) (2025)

## Abstract (arXiv)

Surface parameterization is widely used in computer graphics and geometry processing. It simplifies challenging tasks such as surface registrations, morphing, remeshing and texture mapping. In this paper, we present an efficient algorithm for computing the disk conformal parameterization of simply-connected open surfaces. A double covering technique is used to turn a simply-connected open surface into a genus-0 closed surface, and then a fast algorithm for parameterization of genus-0 closed surfaces can be applied. The symmetry of the double covered surface preserves the efficiency of the computation. A planar parameterization can then be obtained with the aid of a M"obius transformation and the stereographic projection. After that, a normalization step is applied to guarantee the circular boundary. Finally, we achieve a bijective disk conformal parameterization by a composition of quasi-conformal mappings. Experimental results demonstrate a significant improvement in the computational time by over 60%. At the same time, our proposed method retains comparable accuracy, bijectivity and robustness when compared with the state-of-the-art approaches. Applications to texture mapping are presented for illustrating the effectiveness of our proposed algorithm.