An Algorithm for Discrete Constant Mean Curvature Surfaces

Oberknapp, Bernd; Polthier, Konrad

doi:10.1007/978-3-642-59195-2_10

Cited by 22 publications

(11 citation statements)

References 21 publications

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“…Große-Brauckmann and Polthier [19] constructed examples of compact constant mean curvature (CMC) surfaces of low genus numerically, based on a discrete version of the conjugate surface construction [25]. It is an interesting question whether the convergence results of the current paper can help to prove that these numerical examples yield smooth CMC surfaces.…”

Section: Introductionmentioning

confidence: 89%

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On the convergence of metric and geometric properties of polyhedral surfaces

2006

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We provide conditions for convergence of polyhedral surfaces and their discrete geometric properties to smooth surfaces embedded in Euclidean 3-space. Under the assumption of convergence of surfaces in Hausdorff distance, we show that convergence of the following properties are equivalent: surface normals, surface area, metric tensors, and Laplace-Beltrami operators. Additionally, we derive convergence of minimizing geodesics, mean curvature vectors, and solutions to the Dirichlet problem.

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Section: Introductionmentioning

confidence: 89%

“…where C E is the Poincaré constant defined in (25). Since the aspect ratios of the triangles of M n are assumed to be bounded, and u ∈ C ∞ 0 (M) is smooth and supported away from the boundary, ∂M, it follows that the projections u n converge to u, i.e.…”

Section: Discrete Minimal Surfacesmentioning

confidence: 99%

On the convergence of metric and geometric properties of polyhedral surfaces

2006

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“…The mean curvature ( 15,12,7 ) provides a tool to give mathematical existence of these interfaces represents the difference in pressure to proofs of such families. A computer program written by both sides, and a CMC interface minimizes area among sur-Oberknapp ( 16 ) gives a discrete version of this construcfaces with the same enclosed volume, at least on small tion; the program makes use of the special properties of enough subsets.…”

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confidence: 99%

On Gyroid Interfaces

Große-Brauckmann

1997

Journal of Colloid and Interface Science

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“…This can be achieved by solving the Laplace equation ∆ T f = 0 subject to f = 1, where ∆ T f is the tangential component of ∆f on the tangent plane of S 2 . This tangential approach was applied by Oberknapp and Polthier in [64]. Note that this problem is nonlinear because of the constraint f = 1.…”

Section: Harmonic Mapsmentioning

confidence: 99%

Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing

Choi¹,

Ho²,

Lui³

2016

SIAM J. Imaging Sci.

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Abstract. Point cloud is the most fundamental representation of 3D geometric objects. Analyzing and processing point cloud surfaces is important in computer graphics and computer vision. However, most of the existing algorithms for surface analysis require connectivity information. Therefore, it is desirable to develop a mesh structure on point clouds. This task can be simplified with the aid of a parameterization. In particular, conformal parameterizations are advantageous in preserving the geometric information of the point cloud data. In this paper, we extend a state-of-the-art spherical conformal parameterization algorithm for genus-0 closed meshes to the case of point clouds, using an improved approximation of the Laplace-Beltrami operator on data points. Then, we propose an iterative scheme called the North-South reiteration for achieving a spherical conformal parameterization. A balancing scheme is introduced to enhance the distribution of the spherical parameterization. High quality triangulations and quadrangulations can then be built on the point clouds with the aid of the parameterizations. Also, the meshes generated are guaranteed to be genus-0 closed meshes. Moreover, using our proposed spherical conformal parameterization, multilevel representations of point clouds can be easily constructed. Experimental results demonstrate the effectiveness of our proposed framework.Key words. Mesh generation, Triangulation, Quadrangulation, Spherical conformal parameterization, Surface reconstruction, Point cloud, Multilevel representation 1. Introduction. Contemporary scanning technologies enable efficient acquisitions of 3D objects. Using modern 3D scanners, data points are sampled from the surfaces of 3D objects for further analyses and usages. Point clouds are widely applied in computer graphics, vision and many other engineering fields. However, the data points acquired by laser scanners are often complex and unorganized. Moreover, the absence of the connectivity information in point cloud data poses difficulties in understanding the underlying geometry of the 3D objects. This largely hinders the applications of the data. For instance, many applications in 3D printing [38,27] and texture mapping [39,24] are built upon mesh structures. With the rapid development of the computer industry, finding a high quality meshing framework for point cloud data is increasingly important.One possible approach for mesh generation on point clouds is to parameterize a point cloud to a simpler domain with the corresponding genus, such as the unit sphere for genus-0 point clouds. Then, a triangulation or a quadrangulation can be created on the parameter domain instead of the original complicated point cloud. Finally, a mesh structure on the point cloud can be defined with respect to the structure on the parameter domain. The major difficulty of computing parameterizations of point-set surfaces is the extremely limited information they can provide. Most of the existing surface parameterization methods are developed on meshes onl...

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An Algorithm for Discrete Constant Mean Curvature Surfaces

Cited by 22 publications

References 21 publications

On the convergence of metric and geometric properties of polyhedral surfaces

On the convergence of metric and geometric properties of polyhedral surfaces

On Gyroid Interfaces

Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing

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