An analysis and comparison of parameterization-based computation of differential quantities for discrete surfaces

Normals and curvatures are fundamental for geometric modeling and computer-aided design, but their accurate computations on discrete surfaces are challenging. Two types of methods, namely height-function based and parameterization based polynomial fittings, are well founded mathematically and can be proven to deliver convergent results under reasonable assumptions. However, the numerical behaviors of these methods can differ drastically in practice, and no systematic analysis and comparison have been reported previously for these methods. In this paper, we describe a unified framework for these methods based on weighted least squares approximations, and on top of this framework we compare a number of methods in terms of numerical accuracy and stability as well as runtime efficiency and robustness through both theoretical analysis and numerical experiments. Our analysis shows that the choice of parameterization and numerical solver for the least squares problem can have significant impact on the accuracy and stability of polynomial fittings. In addition, we show that the methods based on local orthogonal projection with a safeguard against folding deliver the best combination of simplicity, accuracy, efficiency, and robustness.