# Computational Assessment of Curvatures and Principal Directions of Implicit Surfaces from 3D Scalar Data

@inproceedings{Albin2016ComputationalAO, title={Computational Assessment of Curvatures and Principal Directions of Implicit Surfaces from 3D Scalar Data}, author={Eric Albin and Ronnie Knikker and Shihe Xin and Christian Oliver Paschereit and Yves D’Angelo}, booktitle={MMCS}, year={2016} }

An implicit method based on high-order differentiation to determine the mean, Gaussian and principal curvatures of implicit surfaces from a three-dimensional scalar field is presented and assessed. The method also determines normal vectors and principal directions. Compared to explicit methods, the implicit approach shows robustness and improved accuracy to measure curvatures of implicit surfaces. This is evaluated on simple cases where curvature is known in closed-form. The method is applied…

## 9 Citations

### Methods of arbitrary optimal order with tetrahedral finite-element meshes forming polyhedral approximations of curved domains

- Mathematics
- 2017

Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are demonstrated for a tetrahedron-based Lagrange family of finite elements.

### Ambiguous phase assignment of discretized 3D geometries in topology optimization

- Computer ScienceArXiv
- 2020

### Machine learning algorithms for three-dimensional mean-curvature computation in the level-set method

- Computer ScienceSSRN Electronic Journal
- 2022

The proposed system can yield more accurate mean-curvature estimations in the L 1 and L ∞ norms than modern particle-based interface reconstruction and level-set schemes around under-resolved regions and shows the potential of machine learning to remedy geometrical problems in well-established numerical methods.

### Characterization of surface motion patterns in highly deformable soft tissue organs from dynamic MRI: An application to assess 4D bladder motion

- MedicineComput. Methods Programs Biomed.
- 2022

### Efficient generation of non-cubic stochastic periodic bicontinuous nanoporous structures

- EngineeringComputational Materials Science
- 2019

### Binary Blends of Diblock Copolymers: An Efficient Route to Complex Spherical Packing Phases

- Materials ScienceMacromolecular Theory and Simulations
- 2021

The phase behaviour of binary blends composed of A1B1 and A2B2 diblock copolymers is systematically studied using the polymeric selfconsistent field theory, focusing on the formation and relative…

### Mechanical Properties, Failure Mechanisms, and Scaling Laws of Bicontinuous Nanoporous Metallic Glasses

- Materials ScienceActa Materialia
- 2022

### A criterion for when an emulsion drop undergoing turbulent deformation has reached a critically deformed state

- PhysicsColloids and Surfaces A: Physicochemical and Engineering Aspects
- 2022

## References

SHOWING 1-10 OF 40 REFERENCES

### Estimating Gaussian curvatures from 3D meshes

- Computer ScienceIS&T/SPIE Electronic Imaging
- 2003

The proposed approach can handle vertices with the zero Gaussian curvature uniformly without localizing them as a separate process and the performance is further improved with the local Bezier curve approximation and subdivision.

### Estimating Curvature on Triangular Meshes

- Computer ScienceInt. J. Shape Model.
- 2006

This paper has developed a suite of test cases for assessing both the detailed behavior of these methods, and the error statistics that occur for samples from a general mesh, and provides guidance in choosing an appropriate method for applications requiring curvature estimates.

### Estimating the principal curvatures and the darboux frame from real 3-D range data

- MathematicsIEEE Trans. Syst. Man Cybern. Part B
- 2003

It is shown that with current scanning technology and the algorithms presented here, reliable estimates of the principal curvatures and Darboux frame can be extracted from real data and used in a large variety of tasks.

### A comparison of local surface geometry estimation methods

- Computer ScienceMachine Vision and Applications
- 1997

The nonlinear quadric fitting method considered was found to perform the best but has the greatest computational cost while the facet based approach works as well as the other quadricfitting methods and has a much smaller computational cost.

### Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

- Mathematics, Computer ScienceVisMath
- 2002

A unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes, using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method is proposed.

### An improved geometry-aware curvature discretization for level set methods: Application to tumor growth

- Computer ScienceJ. Comput. Phys.
- 2006

### A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows

- Computer Science
- 1996

Eulerian finite difference methods based on a level set formulation derived for incompressible, immiscible Navier?Stokes equations are proposed and are capable of computing interface singularities such as merging and reconnection.

### A stabilized finite element method using a discontinuous level set approach for the computation of bubble dynamics

- Computer ScienceJ. Comput. Phys.
- 2007

### Curvature formulas for implicit curves and surfaces

- MathematicsComput. Aided Geom. Des.
- 2005