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Curve network reconstruction from a set of unorganized points is an important problem in reverse engineering and computer graphics. In this paper, we propose\ud an automatic method to extract curve segments and reconstruct curve networks from\ud unorganized spatial points. Our proposed method divides reconstruction of curve networks into two steps: 1) detecting nodes of curve segments and 2) reconstructing curve\ud segments. For detection of nodes of curve segments, we present a principal component\ud analysis-based algorithm to obtain candidate nodes from unorganized spatial points\ud and a Euclidean distance-based iterative algorithm to remove peripheral nodes and\ud find the actual nodes. For reconstruction of curve segments, we propose an extraction\ud algorithm to obtain the points on each of curve segments. We present quite a number of examples which use our proposed method to reconstruct curve networks from\ud unorganized spatial points. The results demonstrate the effectiveness of our proposed\ud method and its advantages of good automation and high reconstruction efficiency.

In this paper, we propose a method for constructing spline surfaces interpolating a B-spline curve network, allowing the presence of free parameters, in order to model the interpolating surface. We provide a constructive algorithm for its generation in the case of biquadratic tensor product B-spline surfaces and bivariate B-spline surfaces on criss-cross triangulations. Finally, we present graphical results.

One of the essential problems in the measurement of the freeform surfaces on a coordinate measuring machine is to design appropriate sampling plans to improve the industrial practice in terms of the tradeoff between the sampling accuracy and the efficiency. This paper presents a curve network sampling strategy to approximate the measured surface within a required accuracy while minimizing the cost and time for the measurement by adaptively deriving the optimal sampling locations. The method iteratively extracts two sets of iso-planar curves along two different directions on the parts to form a curve network, which is used to reconstruct the measured surfaces based on the Gordon surface fitting method. Two criteria are integrated to determine the locations of the sampled curves in the sampling process, including the surface complexity and the deviation of the reconstructed surfaces from the CAD model. Both the computer simulation and the actual measurement are conducted to verify the superior sampling efficiency of the proposed method to the conventional raster fashion sampling in measuring freeform surfaces.

Curve network interpolation is an important method for free-form surface modeling. This chapter first discusses the construction of a curve network of arbitrary topology and then introduces a technique for generation of a combined subdivision surface interpolating curve network. Finally, an improved approach to fairing is provided, and two shape modification methods for combined subdivision surfaces are described.

The basic idea of curve network-based design is to construct smoothly connected surface patches, that interpolate boundaries and cross-derivatives extracted from the curve network. While the majority of applications demands only tangent plane G1 continuity between the adjacent patches, curvature continuous connections G2 may also be required. Examples include special curve network configurations with supplemented internal edges, "master-slave" curvature constraints, and general topology surface approximations over meshes.

Recent surface acquisition technologies based on microsensors produce three-space tangential curve data which can be transformed into a network of space curves with surface normals. This paper addresses the problem of surfacing an arbitrary closed 3D curve network with given surface normals. Thanks to the normal vector input, the patch finding problem can be solved unambiguously and an initial piecewise smooth triangle mesh is computed. The input normals are propagated throughout the mesh. Together with the initial mesh, the propagated normals are used to compute mean curvature vectors. We then compute the final mesh as the solution of a new variational optimization method based on the mean curvature vectors. The intuition behind this original approach is to guide the standard Laplacian-based variational methods by the curvature information extracted from the input normals. The normal input increases shape fidelity and allows to achieve globally smooth and visually pleasing shapes. Graphical abstractDisplay Omitted HighlightsA method for surfacing curve networks with prescribed normals is presented.The method does not require an extra parameter to control the magnitude of normals.The key insight is the decoupling of normals from positions before the extrapolation.The resulting shapes are globally smooth and visually pleasing.The method is fully automatic, and, at the same time, simple to implement.

Measuring ultra-precision freeform surfaces with sub-micrometer form accuracy usually requires high density and intensive sampling in order to fully characterize the surface geometry and reduce the measurement uncertainty. However, this usually imposes many challenges such as insufficient sampling or long measurement time. In this paper, a bidirectional curve network based sampling (BCNBS) method is presented for enhancing the performance in measuring ultra-precision freeform surfaces. The BCNBS method is based on scanning two sets of curves on the measured surface in two different directions to form a curve network which is used to construct a substitute surface to represent the measured surface. A series of experiments has been conducted to verify the accuracy and efficiency of proposed sampling method and the results are compared with the conventional one directional raster sampling method. It is interesting to note that the sampling plan produced by the BCNBS method has significant improvement in terms of the efficiency in data sampling and the accuracy in surface geometry representation.

In CAGD the design of a surface that interpolates an arbitrary quadrilateral mesh is definitely a challenging task. The basic requirement is to satisfy both criteria concerning the regularity of the surface and aesthetic concepts.With regard to aesthetic quality, it is well known that interpolatory methods often produce shape artifacts when the data points are unevenly spaced. In the univariate setting, this problem can be overcome, or at least mitigated, by exploiting a proper non-uniform parametrization, that accounts for the geometry of the data. Recently the same principle has been generalized and proven to be effective in the context of bivariate interpolatory subdivision schemes.In this paper, we propose a construction for parametric surfaces of good aesthetic quality and high smoothness interpolating quadrilateral meshes of arbitrary topology. In the classical tensor product setting the same parameter interval must be shared by an entire row or column of mesh edges. Conversely, in this paper, we assign a different parameter interval to each mesh edge. This particular structure, which we call an augmented parametrization, allows us to interpolate each section polyline at parameters values that prevent wiggling of the resulting curve or other interpolation artifacts and yields high quality interpolatory surfaces.The proposed method is generalization of the local univariate spline interpolants introduced in Beccari et al. (2013a) and Antonelli et al. (2014), that have arbitrary continuity and arbitrary order of polynomial reproduction. The generated surfaces retain the same smoothness of the underlying class of univariate splines in the regular regions of the mesh (where, locally, all vertices have valence 4). Mesh regions containing vertices of valence other than 4 are covered with suitably defined surface patches joining the neighboring regular ones with G 1 - or G 2 -continuity. Interpolatory methods often produce shape artifacts for unevenly-spaced data points.A proper non-uniform parametrization can help overcome interpolation artifacts.Parametric surfaces of good aesthetic quality and high smoothness are constructed.The method relies on non-uniform parametrization and locally supported splines.The surfaces interpolate quadrilateral meshes with arbitrary topology.

A simple and effective combined subdivision surface’s modification method is proposed, which is realized by editing its curve network locally. After discussing the overlapping principles of Catmull-Clark subdivision and combined subdivision, a quick update method of the modified surface is given by building the difference curve network from the original curve network and the edited curve network and overlapping the mesh from the difference curve network to the mesh from the original curve network. This method is closely related with the features of combined subdivision, so it is simple and high efficiency. It is useful in the concept design of complex product.

This paper presents a generalized form characterization method named intrinsic feature-based pattern analysis method (IFPAM) for measuring ultra-precision freeform surfaces with sub-micrometer form accuracy. The IFPAM makes use of intrinsic surface properties, the Fourier–Mellin transform, and phase correlation to conduct surface registration. A bidirectional curve network based sampling strategy and a robust surface fitting method are built for accurate representation of the measured freeform surfaces. Compared with traditional least-squares-based methods, the IFPAM not only possesses better robustness and higher precision but also less susceptible to the uncertainty due to geometrical complexity and registration problems involving translation and rotation operations.

Transfinite surface interpolation is a classic topic of computer-aided geometric design (CAGD), and many non-quadrilateral schemes are known. Surfaces defined solely by means of their boundary curves and cross-tangent functions are needed, for example, in three-dimensional curve network-based design, and to fill complex irregular holes such as in vertex blending. This paper deals with interpolating so-called tangential ribbons. Former schemes are enhanced and extended in order to minimize shape artifacts and to provide a more natural patch interior. The proposed representation is based on irregular convex domains that correspond to the lengths and orientations of the boundary curves. The mapping of the individual ribbons within the n -sided domain is calculated by focused parameterization methods that ensure a balanced orientation related to the center of the domain and avoid parametric shearing. Distance-based blending functions ensure that modifying or inserting a small edge will have only a local effect over the n -sided patch. Constructions to create one-sided or two-sided patches are also presented. Examples and open research topics conclude the paper. Highlights? We present three approaches that generalize Coons' patches for n -sided domains. ? Non-regular domain polygons are introduced based on three-dimensional curve networks. ? New distance-based blending functions are proposed to interpolate positional and tangential data. ? We investigate local schemes for optimized surface parameterization.

This paper presents an approach to hull surface transformation based on identifying features in the form topology. The technique allows changes using both interactive mouse manipulation and numerical parameters. A rich transformation tool kit is created which allows the user to confidently make rapid hull changes in conjunction with existing surface definition techniques.

The reconstruction of a surface model from a point cloud is an important task in the reverse engineering of industrial parts. We aim at constructing a curve network on the point cloud that will define the border of the various surface patches. In this paper, we present an algorithm to extract closed sharp feature lines, which is necessary to create such a closed curve network. We use a first order segmentation to extract candidate feature points and process them as a graph to recover the sharp feature lines. To this end, a minimum spanning tree is constructed and afterwards a reconnection procedure closes the lines. The algorithm is fast and gives good results for real-world point sets from industrial applications.

We consider the problem of extremal scattered data interpolation in \(\mathbb {R}^3\). Using our previous work on minimum \(L_2\)-norm interpolation curve networks, we construct a bivariate interpolant \(F\) with the following properties: (i) \(F\) is \(G^1\)-continuous, (ii) \(F\) consists of triangular Bezier surfaces, (iii) each Bezier surface satisfies the tetra-harmonic equation \(\varDelta ^4 F=0\). Hence \(F\) is an extremum to the corresponding energy functional.

Fair webs are energy-minimizing curve networks. Obtained via an extension of cubic splines or splines in tension to networks of curves, they are efficiently computable and possess a variety of interesting applications. We present properties of fair webs and their discrete counterparts, i.e., fair polygon networks. Applications of fair curve and polygon networks include fair surface design and approximation under constraints such as obstacle avoidance or guaranteed error bounds, aesthetic remeshing, parameterization and texture mapping, and surface restoration in geometric models.

International audience; Subdivision surfaces are usually used to construct freeform surfaces from network of curves for its ability and flexibility to deal with complex wireframes. In freeform surface designing, the designers usually draw at first some curves for describing the models conceived in their mind which form a curve network representing an object of arbitrary topology. Then 3D surfaces are computed to interpolate these curves in order to create a B-Rep model. If the subdivision surface is used in the workflow, its control polyhedrons generation from curves polygons could be a time-consuming stage. In this article, we develop an approach to generate automatically a control polyhedral mesh from an arbitrary topological curve network. One of common problems in interpolating surface patch using subdivision surfaces is how to determine the connectivity of control points. Arbitrary topological curve network has no restriction in topology structure, so another problem is that it has more ambiguousness in defining surface patches. There are three steps in our approach. Firstly, we compute a 1D mesh (a unique polygonal model) from curves. Secondly, we identify on the polygon different cycles that would be the boundaries of potential surface patches. Finally, in each identified cycle we apply an algorithm of quadrangulation to construct the control mesh of subdivision.

The footwear industry is a traditional craft sector, where technological advances are difficult to implement owing to the complexity of the processes being carried out, and the level of precision demanded by most of them. The shoe last joining operation is one clear example, where two halves from different lasts are put together, following a specifically traditional process, to create a new one. Existing surface joining techniques analysed in this paper are not well adapted to shoe last design and production processes, which makes their implementation in the industry difficult. This paper presents an alternative surface joining technique, inspired by the traditional work of lastmakers. This way, lastmakers will be able to easily adapt to the new tool and make the most out of their know-how. The technique is based on the use of curve networks that are created on the surfaces to be joined, instead of using discrete data. Finally, a series of joining tests are presented, in which real lasts were successfully joined using a commercial last design software. The method has shown to be valid, efficient, and feasible within the sector.

In this paper we investigate the problem of interpolating a B-spline curve network, in order to create a surface satisfying such a constraint and defined by blending functions spanning the space of bivariate $C^1$ quadratic splines on criss-cross triangulations. We prove the existence and uniqueness of the surface, providing a constructive algorithm for its generation. We also present numerical and graphical results and comparisons with other methods., Comment: With respect to the previous version, this version of the paper is improved. The results have been reorganized and it is more general since it deals with non uniform knot partitions. Accepted for publication in Computer Aided Geometric Design, October 2015

The problem of interpolating a free form curve network with irregular topology is investigated in order to create a curvature continuous surface. The spanning curve segments are parametric quintic polynomials, the interpolating surface elements are biquintic Gregory patches. A necessary compatibility condition is formulated and proved which need to be satisfied at each node of the curve network. Constraints are derived for assuring G 2 continuity between biquintic Gregory patches, which share a common side or a common corner point. The above conditions still leave certain geometric freedom for defining the entire G 2 surface, so following some analysis a particular construction is presented, by which after computing the principle curvatures at each node the free parameters are locally set for each interpolating Gregory patch.

In this paper, we introduce the concept of an exoskeleton as a new abstraction of arbitrary shapes that succinctly conveys both the perceptual and the geometric structure of a 3D model. We extract exoskeletons via a principled framework that combines segmentation and shape approximation. Our method starts from a segmentation of the shape into perceptually relevant parts and then constructs the exoskeleton using a novel extension of the Variational Shape Approximation method. Benefits of the exoskeleton abstraction to graphics applications such as simplification and chartification are presented.

This paper presents a system for designing free-form surfaces starting from a sketched 3D irregular curve network. By simply dragging a smart-pen device in space, the user draws and refines arbitrary 3D style-curves that define an outline of the desired shape. Unlike previous touch-based sketching systems, the user-drawn strokes can both stay on the model surface to reconstruct parts of an existing object, or freely sketch 3D style-lines of non-existing parts to design new geometry. The wireless smart-pen device is supported by an active stereo acquisition system which makes use of two infrared cameras. For a given set of 3D curves, the system automatically constructs a low-resolution mesh that is naturally refined to produce a smooth surface which preserves curvature features defined by the user on the curve network. The interpolating surface is obtained by applying a high-order diffusion flow. We present an efficient two step approach that first diffuses curvature values preserving the curvature constraints, and then corrects the surface to fit the resulting curvature vector field and interpolating the 3D curve network. This leads to fast implementation of a feature preserving fourth order geometric flow. We show several examples to demonstrate the ability of the proposed advanced design methodology to create sophisticated models possibly having sharp creases and corners.

Lofting is a traditional technique for creating a curved shape by first specifying a network of curves that approximates the desired shape and then interpolating these curves with a smooth surface. This paper addresses the problem of lofting from the viewpoint of subdivision. First, we develop a subdivision scheme for an arbitrary network of cubic B-splines capable of being interpolated by a smooth surface. Second, we provide a quadrangulation algorithm to construct the topology of the surface control mesh. Finally, we extend the Catmull-Clark scheme to produce surfaces that interpolate the given curve network. Near the curve network, these lofted subdivision surfaces are C2 bicubic splines, except for those points where three or more curves meet. We prove that the surface is C1 with bounded curvature at these points in the most common cases; empirical results suggest that the surface is also C1 in the general case.

A sketching system for spline-based free-form surfaces on the Responsive Workbench is presented. We propose 3D tools for curve drawing and deformation techniques for curves and surfaces, adapted to the needs of designers. The user directly draws curves in the virtual environment, using a tracked stylus as an input device. A curve network can be formed, describing the skeleton of a virtual model. The non-dominant hand positions and orients the model while the dominant hand uses the editing tools. The curves and the resulting skinning surfaces can interactively be deformed.

This paper proposes a method of generating surfaces from a network of curves that have arbitrary parametric forms, and that intersect in an arbitrary topology.

Recently, along with the improvements of geometry modeling methods using sketch-based interface, there have been a lot of developments in research about generating surface model from 3D curves. However, surfacing a 3D curve network remains an ambiguous problem due to the lack of geometric information. In this paper, we propose a new algorithm for estimating the normal vectors of the 3D curves which accord closely with user intent. Bending energy is defined by utilizing RMF(Rotation-Minimizing Frame) of 3D curve, and we estimated this minimal energy frame as the one that accords design intent. The proposed algorithm is demonstrated with surface model creation of various curve networks. The algorithm of estimating geometric information in 3D curves which is proposed in this paper can be utilized to extract new information in the sketch-based modeling process. Also, a new framework of 3D modeling can be expected through the fusion between curve network and surface creating algorithm.