Given a set of points together with a set of simplices we show how to compute weights associated with the points such that the weighted Delaunay triangulation of the point set contains the simplices, if possible. For a given triangulated surface, this process provides a tetrahedral mesh conforming to the triangulation, i.e. solves the problem of meshing the triangulated surface without inserting additional vertices. The restriction to weighted Delaunay triangulations ensures that the orthogonal dual mesh is embedded, facilitating common geometry processing tasks. We show that the existence of a single simplex in a weighted Delaunay triangulation for given vertices amounts to a set of linear inequalities, one for each vertex. This means that the number of inequalities for a given triangle mesh is quadratic in the number of mesh elements, making the naive approach impractical. We devise an algorithm that incrementally selects a small subset of inequalities, repeatedly updating the weights, until the weighted Delaunay triangulation contains all constrained simplices or the problem becomes infeasible. Applying this algorithm to a range of triangle meshes commonly used graphics demonstrates that many of them admit a conforming weighted Delaunay triangulation, in contrast to conforming or constrained Delaunay that require additional vertices to split the input primitives.

中文翻译：

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符合加权 delaunay 三角剖分

给定一组点和一组单纯形，我们将展示如何计算与点相关的权重，以便点集的加权 Delaunay 三角剖分包含单纯形（如果可能）。对于给定的三角曲面，该过程提供了符合三角剖分的四面体网格，即解决了在不插入额外顶点的情况下对三角曲面进行网格划分的问题。对加权 Delaunay 三角剖分的限制可确保嵌入正交双网格，从而促进常见的几何处理任务。我们表明，对于给定顶点，加权 Delaunay 三角剖分中存在单个单纯形相当于一组线性不等式，每个顶点一个。这意味着给定三角形网格的不等式数量是网格元素数量的二次方，使天真的方法不切实际。我们设计了一种算法，该算法逐步选择不等式的一个小子集，反复更新权重，直到加权 Delaunay 三角剖分包含所有约束单纯形或问题变得不可行。将此算法应用于一系列常用图形的三角形网格表明，它们中的许多都承认符合加权 Delaunay 三角剖分，而符合或约束 Delaunay 则需要额外的顶点来分割输入图元。