Triangular G1 interpolation by 4-splitting domain triangles

Hahmann, Stéfanie; Bonneau, Georges-Pierre

doi:10.1016/s0167-8396(00)00021-2

Cited by 41 publications

(35 citation statements)

References 15 publications

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“…Hahmann and Bonneau [12] presented a piecewise quintic G 1 spline surface interpolating the vertices of a triangular surface mesh of arbitrary topological type. They further improved the method without imposing any constraint on the first derivatives and thus avoid any unwanted undulations when interpolating irregular triangulations [13].…”

Section: Interpolatory Splinementioning

confidence: 99%

A C 1 Globally Interpolatory Spline of Arbitrary Topology

Jin

et al. 2005

Lecture Notes in Computer Science

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Abstract. Converting point samples and/or triangular meshes to a more compact spline representation for arbitrarily topology is both desirable and necessary for computer vision and computer graphics. This paper presents a C 1 manifold interpolatory spline that can exactly pass through all the vertices and interpolate their normals for data input of complicated topological type. Starting from the Powell-Sabin spline as a building block, we integrate the concepts of global parametrization, affine atlas, and splines defined over local, open domains to arrive at an elegant, easy-to-use spline solution for complicated datasets. The proposed global spline scheme enables the rapid surface reconstruction and facilitates the shape editing and analysis functionality.

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Section: Interpolatory Splinementioning

confidence: 99%

A C 1 Globally Interpolatory Spline of Arbitrary Topology

Jin

et al. 2005

Lecture Notes in Computer Science

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“…The methods also attempt to use patches of the lowest possible polynomial degree, sometimes even admitting isolated singular points on patch boundaries; see, e.g., Peters [1991] and Loop [1994]. On the other hand, G 1 methods handle patch configurations of any topological type, requiring no modification for different classes of global topology; see also Hahmann and Bonneau [2000].…”

Section: Alternatives To the Single Planar Domainmentioning

confidence: 99%

A variational method to model G ¹ surfaces over triangular meshes of arbitrary topology in R ³

Sárraga

2000

ACM Trans. Graph.

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This article presents a method for constructing a G 1 -smooth surface, composed of independently parametrized triangular polynomial Bézier patches, to fit scattered data points triangulated in R 3 with arbitrary topology. The method includes a variational technique to optimize the shape of the surface. A systematic development of the method is given, presenting general equations provided by the theory of manifolds, explaining the heuristic assumptions made to simplify calculations, and analyzing the numerical results obtained from fitting two test configurations of scattered data points. The goal of this work is to explore an alternative G 1 construction, inspired by the theory of manifolds, that is subject to fewer application constraints than approaches found in the technical literature; e.g., this approach imposes no artificial restrictions on the tangents of patch boundary curves at vertex points of a G 1 surface. The constructed surface shapes fit all test data surprisingly well for a noniterative method based on polynomial patches.

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“…One alternative solution to the "twist compatibility" problem, or "vertex consistency" problem, was invented by Loop, who designed a scheme using sextic triangular Bézier patches in a one-to-one correspondence with data triangles [Loop 1994]. Hahmann used a similar approach as Loop's scheme to construct the tangents of the boundary curves, then fit four quintic patches to each data triangle to create a G 1 continuous surface [Hahmann and Bonneau 2000]. In all these schemes, control points are set to meet the constraints imposed by having G 1 continuity along the patch boundaries.…”

Section: Introductionmentioning

confidence: 99%

Parametric triangular Bézier surface interpolation with approximate continuity

Liu

Mann

2008

Proceedings of the 2008 ACM Symposium on Solid and Physical Modeling

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A piecewise quintic interpolation scheme with approximate G 1 continuity is presented. For a given triangular mesh of arbitrary topology, one quintic triangular Bézier patch is constructed for each data triangle. Although the resulting surface has G 1 continuity at the data vertices, we only require approximate G 1 continuity along the patch boundaries so as to lower the patch degree. To reduce the normal discontinuity along boundaries, neighbouring patches are adjusted to have identical normals at the middle point of their common boundary. In most cases, the surfaces generated by this scheme have the same level of visual smoothness compared to an existing sextic G 1 continuous interpolation scheme. Further, using the new boundary construction method presented in this paper, better shape quality is observed for sparse data sets than the surfaces of the original G 1 continuous scheme, upon which the new scheme is based.

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Triangular G1 interpolation by 4-splitting domain triangles

Cited by 41 publications

References 15 publications

A C 1 Globally Interpolatory Spline of Arbitrary Topology

A C 1 Globally Interpolatory Spline of Arbitrary Topology

A variational method to model G ¹ surfaces over triangular meshes of arbitrary topology in R ³

Parametric triangular Bézier surface interpolation with approximate continuity

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Triangular G1 interpolation by 4-splitting domain triangles

Cited by 41 publications

References 15 publications

A C 1 Globally Interpolatory Spline of Arbitrary Topology

A C 1 Globally Interpolatory Spline of Arbitrary Topology

A variational method to model G 1 surfaces over triangular meshes of arbitrary topology in R 3

Parametric triangular Bézier surface interpolation with approximate continuity

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Resources

About

A variational method to model G ¹ surfaces over triangular meshes of arbitrary topology in R ³