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自交有理参数曲面网格生成

Mesh Generation for Self-intersecting Rational Parametric Surfaces

  • 摘要: 本文提出一种针对自交有理参数曲面的新型网格生成算法, 它能够在各种度量下生成具有单纯复形结构的三角网格. 本文首先使用动平面法计算曲面的奇异因子, 基于奇异因子本文给出曲面交线在参数域上的隐式方程. 接下来, 本文利用奇异因子和曲面的第一基本形式定位交线上的非一般奇异点, 这些非一般奇异点的精准定位对生成拓扑正确的网格曲面来说至关重要. 然后, 本文设计了一种基于动平面法的交线网格点生成方法, 该方法能保证参数域上生成的网格点包含所有参数曲面交线上网格点的原像. 最后, 本文使用基于粒子的网格生成法生成参数域网格. 大量的实验结果表明本文提出的算法能够保证生成的网格曲面具有复形结构, 并且具有与原参数曲面相同的拓扑结构. 与代表性的最新方法相比, 本文算法在自交参数曲面上具有更优的网格精度和质量.

     

    Abstract: We propose a mesh generation algorithm for rational parametric surfaces with self-intersections that can generate simplicial complex structured triangular meshes with various metrics. Our algorithm first uses the moving plane method to calculate the singular factors of the surface, which provide the implicit equations of the surface intersections in the parameter domain. Next, we locate the non-generic singular points on intersections using the singular factors and the surface's first fundamental form. This step is crucial for generating a mesh surface with correct topology. We then design a point tracing method for the intersections based on the moving plane method, which ensures that the mesh points generated in the parameter domain include all pre-images of the mesh points on the surface intersections. Finally, we use a particle-based mesh generation method to generate the mesh in the parameter domain. We have conducted extensive experiments to validate our algorithm, and the results demonstrate that our algorithm can generate complex structured mesh surfaces with the same topology as the original parametric surfaces. Compared with representative state-of-the-art methods, our algorithm achieves higher mesh accuracy and quality on most self-intersecting parametric surfaces.

     

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