This paper looks at how the choice of planar representation affects spherical parameterization of closed surfaces, and adds a small preconditioner to improve the result.
Sensitivity: Under consistent piecewise-linear assembly, FLASH's unmodified south-pole Beltrami correction reduces exactly to harmonic extension on the source mesh. The authors also give an explicit sensitivity Jacobian and a conditioning bound for finite triangles.
Preconditioner: A bounded five-parameter affine preconditioner (planar scale, translation, anisotropy) is evaluated on the corrected spherical map.
Results: Across 30 genus-0 surfaces, median polar upper-tail log-condition drops by 10.17% and chord-angle distortion by 10.97%, with both improving on every mesh.
Scope: The framework is also applied to conformalized mean-curvature flow and spherical density-equalizing maps.
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This paper looks at how the choice of planar representation affects spherical parameterization of closed surfaces, and adds a small preconditioner to improve the result.