 Award ID(s):
 1762287
 NSFPAR ID:
 10291675
 Date Published:
 Journal Name:
 Computer methods in applied mechanics and engineering
 ISSN:
 18792138
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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null (Ed.)This work discovers the equivalence relation between quadrilateral meshes and meromorphic quartic differentials. Each quadmesh induces a conformal structure of the surface, and a meromorphic quartic differential, where the configuration of singular vertices corresponds to the configurations of the poles and zeros (divisor) of the meromorphic differential. Due to Riemann surface theory, the configuration of singularities of a quadmesh satisfies the Abel–Jacobi condition. Inversely, if a divisor satisfies the Abel–Jacobi condition, then there exists a meromorphic quartic differential whose divisor equals the given one. Furthermore, if the meromorphic quartic differential is with finite trajectories, then it also induces a quadmesh, the poles and zeros of the meromorphic differential correspond to the singular vertices of the quadmesh. Besides the theoretic proofs, the computational algorithm for verification of Abel–Jacobi condition is also explained in detail. Furthermore, constructive algorithm of meromorphic quartic differential on genus zero surfaces is proposed, which is based on the global algebraic representation of meromorphic differentials. Our experimental results demonstrate the efficiency and efficacy of the algorithm. This opens up a novel direction for quadmesh generation using algebraic geometric approach.more » « less

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