Mathematical Foundations of Complex System Science

Numerical Differential Geometry


We have developed a  computational scheme that derives a global polynomial level set parametrisation (GPLS)  for smooth closed surfaces from a regular surface-point set.

The GPLS  enables to approximate a broad class of smooth surfaces by affine algebraic varieties. Posterior, differential-geometric quantities like mean and Gauss curvature can be efficiently and accurately computed. Even 4th-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. Fig.1 shows a  GPLS fit of the prominent Stanford bunny.

We further develop the technique and apply it to a vast class of computational tasks, arising in numerical differential geometry. This includes a High Order Surface Quadrature (HOSQ) technique, capable for integrating highly varying surface integrals, Fig2.


Thekke Veettil, S.K., Zavalani, G., Hernandez Acosta, U., Sbalzarini, I.F., and Hecht, M. Global polynomial level sets for numerical differential geometry of smooth closed surfaces. SIAM J. Sci. Comput., 45(4):A1995-A2018, 2023.
Zavalani, G., Shehu, E., and Hecht. M. A note on the rate of convergence of integration schemes for closed surfaces Comp. Appl. Math. 43, 92 (2024)., 2024, arXiv:2301.02996.
Zavalani, G., Sander, O., and Hecht, M. High-order integration for regular cubical re- parameterizations of triangulated manifolds reaches super-algebraic approximation rates. arXiv:2311.13909