The discrete Laplace‐Beltrami operator for surface meshes is a fundamental building block for many (if not most) geometry processing algorithms. While Laplacians on triangle meshes have been researched intensively, yielding the cotangent discretization as the de‐facto standard, the case of general polygon meshes has received much less attention. We present a discretization of the Laplace operator which is consistent with its expression as the composition of divergence and gradient operators, and is applicable to general polygon meshes, including meshes with non‐convex, and even non‐planar, faces. By virtually inserting a carefully placed point we implicitly refine each polygon into a triangle fan, but then hide the refinement within the matrix assembly. The resulting operator generalizes the cotangent Laplacian, inherits its advantages, and is empirically shown to be on par or even better than the recent polygon Laplacian of Alexa and Wardetzky [AW11] — while being simpler to compute.

中文翻译：

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多边形拉普拉斯算子变得简单

表面网格的离散 Laplace-Beltrami 算子是许多（如果不是大多数）几何处理算法的基本构建块。虽然已经对三角形网格上的拉普拉斯算子进行了深入研究，将余切离散化作为事实上的标准，但一般多边形网格的情况却很少受到关注。我们提出了 Laplace 算子的离散化，这与其作为散度和梯度算子的组合的表达式一致，适用于一般多边形网格，包括具有非凸面甚至非平面面的网格。通过虚拟插入一个精心放置的点，我们隐式地将每个多边形细化为三角形扇形，然后将细化隐藏在矩阵组件中。结果算子推广了余切拉普拉斯算子，继承了它的优点，