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We present a computational scheme that derives a global polynomial level setparametrisation for smooth closed surfaces from a regular surface-point set andprove its uniqueness. This enables us to approximate a broad class of smoothsurfaces by affine algebraic varieties. From such a global polynomial level setparametrisation, differential-geometric quantities like mean and Gausscurvature can be efficiently and accurately computed. Even4$^{\text{th}}$-order terms such as the Laplacian of mean curvature areapproximates with high precision. The accuracy performance results in a gain ofcomputational efficiency, significantly reducing the number of surface pointsrequired compared to classic alternatives that rely on surface meshes orembedding grids. We mathematically derive and empirically demonstrate thestrengths and the limitations of the present approach, suggesting it to beapplicable to a large number of computational tasks in numerical differentialgeometry.

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