- Published 2010 in IEEE Transactions on Visualization and Computer…

This sketch introduces <i>hexagonal global parameterization</i>, a new type of periodic global parameterization that is ideal for tiling surfaces with patterns of six-fold rotational symmetries, i.e., 6-RoSy's [Palacios and Zhang 2007]. Being one of the two most fundamental rotational symmetries that are compatible with translational symmetries in the plane, 6-RoSy's appear in many places in nature, such as honeycombs, insect eyes, corals and crystals, as well as man-made objects such as Islamic patterns [Kaplan and Salesin 2004] and tri-axial weaving [Akleman et al. 2009]. Such symmetries can also provide optimal circle packing, which naturally have applications in architectural design [Schiftner et al. 2009]. A hexagonal global parameterization facilitates all of these applications. See Figure 1 for some examples. Furthermore, parameter lines in a hexagonal global parameterization intersect at an angle of π/3, which enables triangular remeshing of a mesh surface with close to ideal aspect ratios in the triangles. The dual mesh of such a triangulation provides a hexagon-dominant tiling of the surface.

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@article{Nieser2010HexagonalGP,
title={Hexagonal Global Parameterization of Arbitrary Surfaces},
author={Matthias Nieser and Jonathan Palacios and Konrad Polthier and Eugene Zhang},
journal={IEEE Transactions on Visualization and Computer Graphics},
year={2010},
volume={18},
pages={865-878}
}