# An Adaptable Surface Parametrization Method

## Abstract

Parameterizations of triangulated surfaces are used in an increasing number of mesh processing applications for various purposes. Although demands vary, they are often required to preserve the surface metric and thus minimize angle, area and length deformation. However, most of the existing techniques primarily target at angle preservation while disregarding global area deformation.

In this paper an energy functional is proposed, that quantifies angle and global area deformation simultaneously, while the relative importance between angle and area preservation can be controlled by the user through a parameter. We show, how this parameter can be chosen to obtain parameterizations that are optimized for an uniform sampling of the surface of a model. Maps obtained by minimizing this energy are well suited for applications that desire an uniform surface sampling, like re-meshing or mapping regularly patterned textures.

Besides being invariant under rotation and translation of the domain, the energy is designed to prevent face flips during minimization and does not require a fixed boundary in the parameter domain. Although the energy is non-linear, we show how it can be minimized efficiently using non-linear conjugate gradient methods in a hierarchical optimization framework and prove the convergence of the algorithm.

The ability to control the tradeoff between the degree of angle and global area preservation is demonstrated for several models of varying complexity.

Keywords: metric, parameterization, re-meshing, uniform sampling

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## Bibtex

@INPROCEEDINGS{degener-2003-adaptable, author = {Degener, Patrick and Meseth, Jan and Klein, Reinhard}, title = {An Adaptable Surface Parametrization Method}, booktitle = {The 12th International Meshing Roundtable 2003}, year = {2003}, month = sep, keywords = {metric, parameterization, re-meshing, uniform sampling}, abstract = {Parameterizations of triangulated surfaces are used in an increasing number of mesh processing applications for various purposes. Although demands vary, they are often required to preserve the surface metric and thus minimize angle, area and length deformation. However, most of the existing techniques primarily target at angle preservation while disregarding global area deformation. In this paper an energy functional is proposed, that quantifies angle and global area deformation simultaneously, while the relative importance between angle and area preservation can be controlled by the user through a parameter. We show, how this parameter can be chosen to obtain parameterizations that are optimized for an uniform sampling of the surface of a model. Maps obtained by minimizing this energy are well suited for applications that desire an uniform surface sampling, like re-meshing or mapping regularly patterned textures. Besides being invariant under rotation and translation of the domain, the energy is designed to prevent face flips during minimization and does not require a fixed boundary in the parameter domain. Although the energy is non-linear, we show how it can be minimized efficiently using non-linear conjugate gradient methods in a hierarchical optimization framework and prove the convergence of the algorithm. The ability to control the tradeoff between the degree of angle and global area preservation is demonstrated for several models of varying complexity.}, conference = {The 12th International Meshing Roundtable 2003} }