• DOI:10.1137/130939183
• Corpus ID: 15201762

Computing Surface Uniformization Using Discrete Beltrami Flow

@article{Wong2015ComputingSU,
  title={Computing Surface Uniformization Using Discrete Beltrami Flow},
  author={Tsz Wai Wong and Hongkai Zhao},
  journal={SIAM J. Sci. Comput.},
  year={2015},
  volume={37}
}

• T. W. Wong, Hongkai Zhao
• Published 2 June 2015
• Computer Science, Mathematics
• SIAM J. Sci. Comput.

In this paper, we propose a novel algorithm for computing surface uniformization for surfaces with arbitrary topology. According to the celebrated uniformization theorem, all Riemann surfaces can be classified as elliptic, parabolic, or hyperbolic. Our algorithm is able to work on all these cases by first constructing an initial map onto an appropriate domain, such as a sphere, or a polygon in the plane $\mathbb{R}^2$ or the hyperbolic disk $D$, and then morphing the diffeomorphism based on the discrete Beltrami flow algorithm. For high genus surfaces, both the final mapping and the target domain are unknown, which presents a challenge in general. Each such surface can be conformally mapped onto $D$ modulo a discrete subgroup of all fractional linear transforms on $D$. A conformal copy of the surface, which is also its uniformization domain, can be visualized as a fundamental polygon in $D$ corresponding to the discrete subgroup, where the generators of the discrete subgroup map each side of the fundament...