Optimal Moebius Transformations for Information Visualization and Meshing

TL;DR

This paper introduces linear-time algorithms for optimizing Moebius transformations to enhance various applications like spherical graph drawing, hyperbolic browsing, mesh generation, and brain mapping by maximizing minimal distances or sizes.

Contribution

It presents novel quasiconvex programming algorithms for efficiently finding optimal Moebius transformations in different geometric contexts.

Findings

Algorithms run in linear time

Effective in maximizing minimal sphere size

Applicable to multiple visualization and meshing problems

Abstract

We give linear-time quasiconvex programming algorithms for finding a Moebius transformation of a set of spheres in a unit ball or on the surface of a unit sphere that maximizes the minimum size of a transformed sphere. We can also use similar methods to maximize the minimum distance among a set of pairs of input points. We apply these results to vertex separation and symmetry display in spherical graph drawing, viewpoint selection in hyperbolic browsing, element size control in conformal structured mesh generation, and brain flat mapping.