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Triangular G1 interpolation by 4splitting domain triangles, Computer Aided Geometric Design 17
, 2000
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Rapport de Recherche LMCIMAG
Smoothing Polyhedra Made Easy
, 1995
"... A mesh of points outlining a surface is polyhedral if all cells are either quadrilateral or planar. A mesh is vertexdegree bounded, if at most four cells meet at every vertex. This paper shows that if a mesh has both properties then simple averaging of its points yields the BernsteinB'ezier c ..."
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Cited by 5 (3 self)
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A mesh of points outlining a surface is polyhedral if all cells are either quadrilateral or planar. A mesh is vertexdegree bounded, if at most four cells meet at every vertex. This paper shows that if a mesh has both properties then simple averaging of its points yields the BernsteinB'ezier coefficients of a smooth, at most cubic surface that consists of twice as many threesided polynomial pieces as there are interior edges in the mesh. Meshes with checker board structure, i.e. rectilinear meshes are a special case and result in a quadratic surface. Since any mesh and, in particular any wireframe of a polyhedron can be refined, by averaging, to a vertexdegree bounded polyhedral mesh this result allows reinterpreting a number of algorithms that construct smooth surfaces and advertises the corresponding averaging formulas as a basis for a wider class of algorithms.
Fair Surface Reconstruction Using Quadratic Functionals
, 1995
"... An algorithm for surface reconstruction from a polyhedron with arbitrary topology consisting of triangular faces is presented. The first variant of the algorithm constructs a curve network consisting of cubic B'ezier curves meeting with tangent plane continuity at the vertices. This curve netwo ..."
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Cited by 4 (0 self)
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An algorithm for surface reconstruction from a polyhedron with arbitrary topology consisting of triangular faces is presented. The first variant of the algorithm constructs a curve network consisting of cubic B'ezier curves meeting with tangent plane continuity at the vertices. This curve network is extended to a smooth surface by replacing each of the networks facets with a split patch consisting of three triangular B'ezier patches. The remaining degrees of freedom of the curve network and the split patches are determined by minimizing a quadratic functional. This optimization process works either for the curve network and the split patches separately or in one simultaneous step. The second variant of our algorithm is based on the construction of an optimized curve network with higher continuity. Examples demonstrate the quality of the different methods. 1 Introduction The reconstruction of a surface from a set of (a priori unorganized) points as well as the design of surfaces with a...
Triangular G¹ Interpolation by 4Splitting Domain Triangles
, 1999
"... A piecewise quintic G¹ spline surface interpolating the vertices of a triangular surface mesh of arbitrary topological type is presented. The surface has an explicit triangular B'ezier representation, is affine invariant and has local support. The twist compatibility problem which arises when j ..."
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Cited by 3 (3 self)
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A piecewise quintic G¹ spline surface interpolating the vertices of a triangular surface mesh of arbitrary topological type is presented. The surface has an explicit triangular B'ezier representation, is affine invariant and has local support. The twist compatibility problem which arises when joining an even number of polynomial patches G¹ continuously around a common vertex is solved by constructing C²consistent boundary curves. Piecewise C¹ boundary curves and a regular 4split of the domain triangle make shape parameters available for controlling locally the boundary curves. A small number of free inner control points can be chosen for some additional local shape effects.
Abstract Approximate Continuity for Parametric Bézier Patches
"... In this paper, we present a piecewise cubic, parametric surface scheme to interpolate positions and normals on a triangulated data set. For each data triangle, we fit three triangular cubic patches in a CloughTocher like arrangement. However, while we construct the micropatches to meet each other ..."
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In this paper, we present a piecewise cubic, parametric surface scheme to interpolate positions and normals on a triangulated data set. For each data triangle, we fit three triangular cubic patches in a CloughTocher like arrangement. However, while we construct the micropatches to meet each other C 1, we only require approximate G 1 continuity across macropatches boundaries. To control the normal discontinuity on the macropatch boundaries, neighbouring patches are constructed to interpolate the position and normals at the ends of their common boundary, as well as to have equal normals at additional points on the boundary. The resulting scheme constructs patches with similar shape to the quartic ShirmanSéquin construction, and has better shape than Peters ’ G 1 cubic scheme on near singular data.
Using Local Optimization in Surface Fitting
, 1995
"... . Local optimization is used to set the free parameters in a triangular surface fitting scheme, resulting in surfaces with better shape. While some of the free parameters can be set to match curvature information, other free parameters are independent of this information. x1. Introduction A large n ..."
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. Local optimization is used to set the free parameters in a triangular surface fitting scheme, resulting in surfaces with better shape. While some of the free parameters can be set to match curvature information, other free parameters are independent of this information. x1. Introduction A large number of local parametric triangular surface schemes have been developed over the past fifteen years (see [8,9] for a survey of such schemes). These schemes are local in that changes to part of the data only affect portions of the surface near the changed data. Surprisingly, all of these schemes exhibit similar shape defects. On closer inspection, it is seen that these schemes all have a large number of free parameters that are set using simple heuristics. By manually adjusting these parameters, one can improve the shape of the surfaces [9]. In this paper, I will investigate using local optimization to set the free parameters in a local, triangular, splitdomain, polynomial surface interpola...
Surface Reconstruction Based Upon
"... In this paper we present a method for surface reconstruction using an extension of the Minimum Norm Network (MNN) designed for interpolating (functional) scattered data to the parametric case. Given a polyhedron with triangular faces, we obtain a smooth surface interpolating the vertices of the ..."
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In this paper we present a method for surface reconstruction using an extension of the Minimum Norm Network (MNN) designed for interpolating (functional) scattered data to the parametric case. Given a polyhedron with triangular faces, we obtain a smooth surface interpolating the vertices of the polyhedron and preserving its topology. As for functional MNN, a curve network is constructed satisfying certain conditions at the vertices from where the curves emanate, and having minimal norm with respect to a certain functional. The G MNN can then be extended to a smooth surface using, for example, methods described by Shirman/Sequin, Peters and Mann. Additionally we give a construction for a G MNN and describe its extension to a smooth surface.
Approximate Continuity for Functional, Triangular
"... In this paper, we investigate a relaxation of the C1 continuity conditions between functional, triangular Bézier patches, allowing for patches to meet approximately C1. We analyze the cross boundary continuity of functional triangular Bézier patches, and derive a bound for the discontinuity in the n ..."
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In this paper, we investigate a relaxation of the C1 continuity conditions between functional, triangular Bézier patches, allowing for patches to meet approximately C1. We analyze the cross boundary continuity of functional triangular Bézier patches, and derive a bound for the discontinuity in the normals between two patches based on their control points. We test our discontinuity bound on a simple data fitting scheme using cubic patches. 1
Parametric Triangular Bézier Surface Interpolation with Approximate Continuity
"... 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 ..."
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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 than the surfaces of the original G 1 continuous scheme, upon which the new scheme is based.
APPROXIMATE G¹ CUBIC SURFACES FOR DATA APPROXIMATION
"... This paper presents a piecewise cubic approximation method with approximate G¹ continuity. For a given triangular mesh of points with arbitrary topology, one cubic triangular Bézier patch surface is constructed. The resulting surfaces have G¹ continuity at the vertex points, but only requires approx ..."
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This paper presents a piecewise cubic approximation method with approximate G¹ continuity. For a given triangular mesh of points with arbitrary topology, one cubic triangular Bézier patch surface is constructed. The resulting surfaces have G¹ continuity at the vertex points, but only requires approximate G¹ continuity along the macropatch boundaries so as to lower the patch degree. While our scheme cannot generate the surfaces in as high quality as Loop’s sextic scheme, they are of half the polynomial degree, and of far better shape quality than the results of interpolating split domain schemes.