基于强加奇异位形的可积交叉场生成--二维流形情形--
Integrable cross-field generation based on imposed singularity configuration -- the 2D manifold case --
摘要：
本文提出了基于用户施加的奇异性配置在二维流形上生成可积交叉场的数学基础。在本文中，我们要么使用自然出现的奇点，例如，通过求解非线性问题，或者使用用户定义的奇点模式作为输入，可能具有通常不会出现在交叉场计算中的高价奇点。这个奇异集是在Abel-Jacobi条件的约束下有效的奇异位形。该文件的主要贡献是发展的一个配方，允许计算一个可积的各向同性的2D交叉场从一组给定的奇点通过分辨率只有两个线性偏微分方程。为了解决可能的次优奇点分布的问题，我们还提出了一个可积的各向异性的2D交叉场的生成基于用户施加的奇异模式的数学设置。开发的配方支持各向同性和各向异性块结构的四边形网格生成。关键词：可积二维交叉场，有效奇异位形，四边形布局，四边形网格划分
英文摘要：
This work presents the mathematical foundations for the generation of integrable cross-field on 2D manifolds based on user-imposed singularity configuration. In this paper, we either use singularities that appear naturally, e.g., by solving a non-linear problem, or use as an input user-defined singularity pattern, possibly with high valence singularities that typically do not appear in cross-field computations. This singularity set is under the constraint of Abel-Jacobi's conditions for valid singularity configurations. The main contribution of the paper is the development of a formulation that allows computing an integrable isotropic 2D cross-field from a given set of singularities through the resolution of only two linear PDEs. To address the issue of possible suboptimal singularities' distribution, we also present the mathematical setting for the generation of an integrable anisotropic 2D cross-field based on a user-imposed singularity pattern. The developed formulations support both an isotropic and an anisotropic block-structured quad mesh generation. Keywords: integrable 2D cross-field, valid singularity configuration, quad layout, quad meshing
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