On surface normal and Gaussian curvature approximations given data sampled from a smooth surface
Computer Aided Geometric Design
A survey of methods for recovering quadrics in triangle meshes
ACM Computing Surveys (CSUR)
Least squares conformal maps for automatic texture atlas generation
Proceedings of the 29th annual conference on Computer graphics and interactive techniques
Computer Aided Geometric Design
Restricted delaunay triangulations and normal cycle
Proceedings of the nineteenth annual symposium on Computational geometry
Estimating the tensor of curvature of a surface from a polyhedral approximation
ICCV '95 Proceedings of the Fifth International Conference on Computer Vision
Fundamentals of spherical parameterization for 3D meshes
ACM SIGGRAPH 2003 Papers
On the angular defect of triangulations and the pointwise approximation of curvatures
Computer Aided Geometric Design
A novel cubic-order algorithm for approximating principal direction vectors
ACM Transactions on Graphics (TOG)
Estimating Curvatures and Their Derivatives on Triangle Meshes
3DPVT '04 Proceedings of the 3D Data Processing, Visualization, and Transmission, 2nd International Symposium
Estimating differential quantities using polynomial fitting of osculating jets
Computer Aided Geometric Design
Exact and interpolatory quadratures for curvature tensor estimation
Computer Aided Geometric Design
Consistent computation of first- and second-order differential quantities for surface meshes
Proceedings of the 2008 ACM symposium on Solid and physical modeling
ACM Transactions on Mathematical Software (TOMS)
Convergence of discrete Laplace-Beltrami operators over surfaces
Computers & Mathematics with Applications
On the curvature effect of thin membranes
Journal of Computational Physics
Curvature tensor computation by piecewise surface interpolation
Computer-Aided Design
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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.