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Geometric optimization using nonlinear rotation-invariant coordinates

Josua Sassen, Behrend Heeren, Klaus Hildebrandt, and Martin Rumpf

Left: Input shapes X1 and X2 (taken from [PRMB15]) and reconstruction of linear average (Z(X1) +Z(X2))/2 ∈ M/ with the local violations of the integrability condition as color map. Rightmost shapes: reconstruction using various spanning trees color coded with respect to the order of traversal.

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We consider Nonlinear Rotation-Invariant Coordinates (NRIC) representing triangle meshes with fixed combinatorics as a vector stacking all edge lengths and dihedral angles. Previously, conditions for the existence of vertex positions matching given NRIC have been established. We develop the machinery needed to use NRIC for solving geometric optimization problems. Moreover, we introduce a fast and robust algorithm that reconstructs vertex positions from close-to integrable NRIC. Our experiments underline that NRIC-based optimization is especially effective for near-isometric problems.

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Citation

Josua Sassen, Behrend Heeren, Klaus Hildebrandt, and Martin Rumpf, Geometric optimization using nonlinear rotation-invariant coordinates, CoRR, abs/1908.11728, p. 101829, 2019.

BibTex 

@article{bib:sassen:2019,
    author       = { Sassen, Josua and Heeren, Behrend and Hildebrandt, Klaus and Rumpf, Martin },    
    title        = { Geometric optimization using nonlinear rotation-invariant coordinates },
    journal      = { CoRR },
    volume       = { abs/1908.11728 },
    year         = { 2019 },
    pages        = { 101829 },
    doi          = { 10.1016/j.cagd.2020.101829 },
    dblp         = { journals/corr/abs-1908-11728 },
    url          = { https://publications.graphics.tudelft.nl/papers/108 },
}