Geometric Accuracy Analysis for Discrete Surface Approximation

Dai, Jiajun; Luo, Wei; Yau, Shing–Tung; Gu, Xianfeng

doi:10.1007/11802914_5

Cited by 7 publications

(9 citation statements)

References 23 publications

(25 reference statements)

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“…However, we have not demonstrated that the new sampling conditions are sufficient to ensure that the iDt‐mesh is a substructure of the 3D Delaunay tetrahedralization. Since good convergence properties of the iDt‐mesh have been demonstrated [DLYG06], this would put it on a more or less equal footing with the rDt. It is known that the rDt and the iDt‐mesh are not necessarily combinatorially equivalent, regardless of sampling density [DZM07].…”

Section: Discussionmentioning

confidence: 99%

“…Although our work focuses more on the qualitative local behaviour of geodesics, we did exploit these works when it was necessary to bound geodesic lengths to allow a comparison between intrinsic and extrinsic sampling criteria. Dai et al [DLYG06] have also recently produced a geometric accuracy analysis. Their analysis is from the intrinsic viewpoint, using Leibon and Letscher's work [LL00] as the topological correctness foundation.…”

Section: Introductionmentioning

confidence: 99%

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Surface sampling and the intrinsic Voronoi diagram

Dyer¹,

Zhang²,

Möller³

2008

Computer Graphics Forum

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We develop adaptive sampling criteria which guarantee a topologically faithful mesh and demonstrate an improvement and simplification over earlier results, albeit restricted to 2D surfaces. These sampling criteria are based on functions defined by intrinsic properties of the surface: the strong convexity radius and the injectivity radius. We establish inequalities that relate these functions to the local feature size, thus enabling a comparison between the demands of the intrinsic sampling criteria and those based on Euclidean distances and the medial axis.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Surface sampling and the intrinsic Voronoi diagram

Dyer¹,

Zhang²,

Möller³

2008

Computer Graphics Forum

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“…The concepts of conformal maps, hyperbolic metrics and Ricci flow are also translated from the smooth surface category to the discrete mesh category. We show the Riemannian metrics of meshes induced by Delaunay triangulations converge to the metric on the smooth surface in [Dai et al 2006]. The convergence of conformal structures is proved in [Luo 2006].…”

Section: Hyperbolic Discrete Variational Ricci Flowmentioning

confidence: 98%

Computing surface hyperbolic structure and real projective structure

Jin

Luo

2006

Proceedings of the 2006 ACM Symposium on Solid and Physical Modeling

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Figure 1: The hyperbolic structure and real projective structure of a genus two closed surface. From left to right, the original surface with a set of canonical fundamental group basis; the isometric embedding of its universal covering space in the Poincaré disk with the hyperbolic uniformization metric; the isometric embedding in the Klein disk which induces the real projective structure; the isometric embedding in the Poincaré disk, where the boundaries of fundamental domains are straightened to hyperbolic lines. AbstractGeometric structures are natural structures of surfaces, which enable different geometries to be defined on the surfaces. Algorithms designed for planar domains based on a specific geometry can be systematically generalized to surface domains via the corresponding geometric structure. For example, polar form splines with planar domains are based on affine invariants. Polar form splines can be generalized to manifold splines on the surfaces which admit affine structures and are equipped with affine geometries.Surfaces with negative Euler characteristic numbers admit hyperbolic structures and allow hyperbolic geometry. All surfaces admit real projective structures and are equipped with real projective geometry. Because of their general existence, both hyperbolic structures and real projective structures have the potential to replace the role of affine structures in defining manifold splines. This paper introduces theoretically rigorous and practically simple algorithms to compute hyperbolic structures and real projective structures for general surfaces. The method is based on a novel geometric tool -discrete variational Ricci flow. Any metric surface admits a special uniformization metric, which is conformal to its original metric and induces constant curvature. Ricci flow is an efficient method to calculate the uniformization metric, which determines the hyperbolic structure and real projective structure.The algorithms have been verified on real surfaces scanned from sculptures. The method is efficient and robust in practice. To the best of our knowledge, this is the first work of introducing algorithms based on Ricci flow to compute hyperbolic structure and real projective structure. * More importantly, this work introduces the framework of general geometric structures, which enable different geometries to be defined on manifolds and lay down the theoretical foundation for many important applications in geometric modeling.

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“…Under the assumption of convergence of surfaces in Hausdorff distance, Hildebrandt et al [12] proved that convergence of the following properties are equivalent: surface normals, surface area, metric tensors, and Laplace Beltrami operators. Dai et al [7] derived the explicit formulae to the bounds of Hausdorff distance, normal distance, and Riemannian metric distortion between the smooth surface and the triangle mesh. They proved that the meshes induced from Delaunay triangulations of dense samples on a smooth surface are convergent to the smooth surface under both Hausdorff distance and normal fields.…”

Section: Related Workmentioning

confidence: 99%