论文标题
平滑封闭表面的数值微分几何形状的全局多项式水平集
Global Polynomial Level Sets for Numerical Differential Geometry of Smooth Closed Surfaces
论文作者
论文摘要
我们提出了一种计算方案，该方案从常规的表面点集中衍生出平滑闭合表面的全局多项式级集参数化，并证明其独特性。这使我们能够通过仿射代数品种近似一类光滑的表面。从这样的全局多项式设定参数化，可以有效且准确地计算差分几何数量（如均值和高斯曲率）。即使是4 $^{\ text {th}} $ - 订购术语，例如平均曲率的拉普拉斯式的近似值是高精度。与依赖于表面网格或嵌入网格的经典替代方案相比，准确性性能会导致计算效率的提高，从而显着减少了所需的表面点数。我们从数学上得出并在经验上证明了本方法的优势和局限性，这表明它适用于数值差异几何形状中的大量计算任务。
We present a computational scheme that derives a global polynomial level set parametrisation for smooth closed surfaces from a regular surface-point set and prove its uniqueness. This enables us to approximate a broad class of smooth surfaces by affine algebraic varieties. From such a global polynomial level set parametrisation, differential-geometric quantities like mean and Gauss curvature can be efficiently and accurately computed. Even 4$^{\text{th}}$-order terms such as the Laplacian of mean curvature are approximates with high precision. The accuracy performance results in a gain of computational efficiency, significantly reducing the number of surface points required compared to classic alternatives that rely on surface meshes or embedding grids. We mathematically derive and empirically demonstrate the strengths and the limitations of the present approach, suggesting it to be applicable to a large number of computational tasks in numerical differential geometry.
