GLOBAL POLYNOMIAL LEVEL SETS FOR NUMERICAL DIFFERENTIAL GEOMETRY OF SMOOTH CLOSED SURFACES

We present a computational scheme that derives a global polynomial level set parameterization 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 parameterization, differential-geometric quantities like mean and Gauss curvature can be efficiently and accurately computed. Even fourth-order terms such as the Laplacian of mean curvature are approximated 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.

Automated summary

We propose a computational scheme that approximates smooth closed surfaces by deriving a global polynomial level set parameterization from a regular surface-point set, and prove its uniqueness. This method efficiently and accurately computes differential-geometric quantities and has high precision for approximating fourth-order terms. It outperforms classic alternatives in terms of computational efficiency.