Matrix computations (3rd ed.)
Curves and surfaces in geometric modeling: theory and algorithms
Curves and surfaces in geometric modeling: theory and algorithms
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
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)
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
A novel cubic-order algorithm for approximating principal direction vectors
ACM Transactions on Graphics (TOG)
Ridge-valley lines on meshes via implicit surface fitting
ACM SIGGRAPH 2004 Papers
Estimating Curvatures and Their Derivatives on Triangle Meshes
3DPVT '04 Proceedings of the 3D Data Processing, Visualization, and Transmission, 2nd International Symposium
Normal Based Estimation of the Curvature Tensor for Triangular Meshes
PG '04 Proceedings of the Computer Graphics and Applications, 12th Pacific Conference
Estimating differential quantities using polynomial fitting of osculating jets
Computer Aided Geometric Design
A Sampling Framework for Accurate Curvature Estimation in Discrete Surfaces
IEEE Transactions on Visualization and Computer Graphics
Curvature-based Energy for Simulation and Variational Modeling
SMI '05 Proceedings of the International Conference on Shape Modeling and Applications 2005
Robust principal curvatures on multiple scales
SGP '06 Proceedings of the fourth Eurographics symposium on Geometry processing
Curvature estimation scheme for triangle meshes using biquadratic Bézier patches
Computer-Aided Design
Convergence of discrete Laplace-Beltrami operators over surfaces
Computers & Mathematics with Applications
Computer Aided Geometric Design
Robust principal curvatures using feature adapted integral invariants
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
Local Orthogonal Cutting Method for Computing Medial Curves and Its Biomedical Applications
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
On the curvature effect of thin membranes
Journal of Computational Physics
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Differential quantities, including normals, curvatures, principal directions, and associated matrices, play a fundamental role in geometric processing and physics-based modeling. Computing these differential quantities consistently on surface meshes is important and challenging, and some existing methods often produce inconsistent results and require ad hoc fixes. In this paper, we show that the computation of the gradient and Hessian of a height function provides the foundation for consistently computing the differential quantities. We derive simple, explicit formulas for the transformations between the first- and second-order differential quantities (i.e., normal vector and curvature matrix) of a smooth surface and the first- and second-order derivatives (i.e., gradient and Hessian) of its corresponding height function. We then investigate a general, flexible numerical framework to estimate the derivatives of the height function based on local polynomial fittings formulated as weighted least squares approximations. We also propose an iterative fitting scheme to improve accuracy. This framework generalizes polynomial fitting and addresses some of its accuracy and stability issues, as demonstrated by our theoretical analysis as well as experimental results.