Global Polynomial Level Sets for Numerical Differential Geometry of Smooth Closed Surfaces

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.