  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.
