On increasing the developability of a trimmed NURBS surface

Developable surfaces are desired in designing products manufactured from planar sheets. Trimmed non-uniform rational B-spline (NURBS) surface patches are widely adopted to represent 3D products in CAD/CAM. This paper presents a new method to increase the developability of an arbitrarily trimmed NURBS surface patch. With this tool, designers can first create and modify the shape of a product without thinking about the developable constraint. When the design is finished, our approach is applied to increase the developability of the designed surface patches. Our method is an optimisation-based approach. After defining a function to identify the developability of a surface patch, the objective function for increasing the developability is derived. During the optimisation, the positions and weights of the free control points are adjusted. When increasing the developability of a given surface patch, its deformation is also minimised and the singular points are avoided. G⁰ continuity is reserved on the boundary curves during the optimisetion, and the method to reserve G¹ continuity across the boundaries is also discussed in this paper. Compared to other existing methods, our approach solves the problem in a novel way that is close to the design convention, and we are dealing with the developability problem of an arbitrarily trimmed NURBS patch.     

Filling n-sided regions with G 1 triangular Coons B-spline patches

Filling n-sided regions is an essential operation in shape and surface modeling. Positional and tangential continuities are highly required in designing and manufacturing. We propose a method for filling n-sided regions with untrimmed triangular Coons B-spline patches, preserving G ¹ continuity exactly. The algorithm first computes a central point, a central normal, the central, and the corner derivative vectors. Then the region is split into n triangular areas by connecting the central point to each corner of the boundary. These inner curves and all cross-boundary derivatives are computed fulfilling G ¹ compatibility conditions. And finally, the triangular patches are generated in the Coons B-spline form, one boundary of which is regressed to the central vertex. Neither positional nor tangential error is introduced by this method. And only one degree elevation is needed.     

基于多分辨率模型的三角曲面特征线辨识技术

由于特征线在反求工程 CAD建模中具有非常重要的作用 ,因此利用图形图象处理中的多分辨率模型概念 ,通过研究三角曲面模型的特征线 ,提出了一种三角曲面特征线的计算方法 .将计算得到的初始特征线通过编辑、修改等手段进行处理 ,得到清晰的特征线 ,并将其作为进一步划分重构 B样条曲面边界的依据和参考 ,从而为实现基于三角曲面模型的 B样条曲面重构奠定了基础 .实验结果证明 ,该算法能够在三角曲面上提取出令人满意的特征线 ,并据此重构出拓扑划分合理的 B样条曲面 .     

Interactive sketch generation

In this paper, we propose an interactive system for generating artistic sketches from images, based on the stylized multiresolution B-spline curve model and the livewire contour tracing paradigm. Our multiresolution B-spline stroke model allows interactive and continuous control of style and shape of the stroke at any level of details. Especially, we introduce a novel mathematical paradigm called the wavelet frame to provide essential properties for multiresolution stroke editing, such as feature point preservation, locality, time-efficiency, good approximation, etc. The livewire stroke map construction leads the user-guided stroke to automatically lock on to the target contour, allowing fast and accurate sketch drawing. We classify the target contours as outlines and interior flow, and develop two respective livewire techniques based on extended graph formulation and vector flow field. Experimental results show that the proposed system facilitates quick and easy generation of artistic sketches of various styles.     

A practical construction of G smooth biquintic B-spline surfaces over arbitrary topology

The motivation of this paper is to develop a local scheme of constructing G¹ smooth B-spline surfaces with single interior knots over arbitrary topology. In this paper, we obtain the conditions of G¹ continuity between two adjacent biquintic B-spline surfaces with interior single knots. These conditions are directly represented by the relevant control points of the two B-spline surfaces. By utilizing these G¹ conditions, we develop the first local scheme of constructing G¹ smooth biquintic B-spline surfaces with interior single knots for arbitrary topological type. The high complexity of deriving the local G¹ scheme is well overwhelmed. The biquintic is the lowest degree for which there exists a local scheme of constructing G¹ smooth B-spline surfaces with interior single knots over arbitrary topology.     

New multiresolution modeling techniques in CAD

To apply wavelet transformation to CAD surface model composed of multiple surface patches, an algorithm called wavelet signal separation to preserve geometric constraints during wavelet transformation is proposed. The algorithm divides the B-spline control points into those associated and those unassociated with the geometric constraints. Through preserving the signal information associated with the constraint control points, the geometric constraints can be automatically preserved after wavelet transformation. This paper also briefly investigates two types of the detail feature propagation technique in wavelet-based multiresolution CAD system. One is detail feature motion. The other is detail feature repetition. Finally, a comprehensive example is presented to illustrate the effects of the combination of those techniques.     

Haptic sculpting of multi-resolution B-spline surfaces with shaped tools

In this paper, we first propose an implicit surface to B-spline surface haptic interface, which provides both force and torque feedback. We then present a new haptic sculpting system for B-spline surfaces with shaped tools of implicit surface. In the physical world, people touch or sculpt with their fingers or tools, instead of just manipulating points. Shaped virtual sculpting tools help users to relate the virtual modeling process to physical-world experience. Various novel haptic sculpting operations are developed to make the sculpting of B-spline surfaces more intuitive. Wavelet-based multi-resolution tools are provided to let modelers adjust the resolution of sculpture surfaces and thus the scale of deformation can be easily controlled. Moreover, sweep editing and 3D texture have been implemented by taking advantage of both the wavelet technique and haptic sculpting tools.     

Generation of Discrete Bicubic G^1 B-Spline Ship Hullform Surfaces from a Given Curve Network Using Virtual Iso-Parametric Curves

We propose a method that automatically generates discrete bicubic G^1 continuous B-spline surfaces that interpolate the curve network of a ship huliform.First,the curves in the network are classified into two types;boundary curves and ＂reference curves＂,The boundary curves correspond to a set of rectangular（or triangular）topological type that can be representes with tensot-product （or degenerate）B-spline surface patches.Next,in the interior of the patches,surface fitting points and cross boundary derivatives are estimated from the reference curves by constructing ＂virtual＂isoparametric curves.Finally,a discrete G^1 continuous B-spline surface is gencrated by a surface fitting algorithm.Several smooth ship hullform surfaces generated from curve networks corresponding to actual ship hullforms demonstrate the quality of the method.     

Interpolating G Bézier surfaces over irregular curve networks for ship hull design

We propose a local method of constructing piecewise G¹ Bézier patches to span an irregular curve network, without modifying the given curves at odd- and 4-valent node points. Topologically irregular regions of the network are approximated by implicit surfaces, which are used to generate split curves, which subdivide the regions into triangular and/or rectangular sub-regions. The subdivided regions are then interpolated with Bézier patches. We analyze various singular cases of the G¹ condition that is to be met by the interpolation and propose a new G¹ continuity condition using linear and quartic scalar weight functions. Using this condition, a curve network can be interpolated without modification at 4-valent nodes with two collinear tangent vectors, even in the presence of singularities. We demonstrate our approach in a ship hull.     

Multiresolution for Closed Curves and Surfaces Using Wavelets

Multiresolution modeling is becoming a powerful tool for fast display, and geometric data compression and transmission of complex shapes. Most of the existing literatures investigating the multiresolution for B-spline curves and surfaces are concentrated on open ones. In this paper, we focus on the multiresolution representations and editing of closed B-spline curves and surfaces using wavelets. A repetition approach is adopted for the multiresolution analysis of closed B-spline curves and surfaces. Since the closed curve or surface itself is periodic, it can overcome the drawback brought by the repetition method, i.e. introducing the discontinuities at the boundaries. Based on the models at different multiresolution levels, the multiresolution editing methods of closed curves and surfaces are introduced. Users can edit the overall shape of a closed one while preserving its details, or change its details without affecting its overall shape.     

A surface modeling method based on the envelope template

In this paper, we present a novel surface modeling scheme based on an envelope template. A two-parameter family of interpolating surfaces is generated by repeated bicubic interpolation of the given data points, and then a solution to the envelope condition and the envelope of the family are constructed. The continuity conditions of two adjacent patches along the common boundary are derived by analyzing the geometric properties of the envelope patch. In order to facilitate surface modeling, an envelope template is constructed, which has many desirable advantages including simple structure, good local features and so on. G² or C² composite surfaces can be obtained utilizing the envelope template sweeping over the data points.     

对可调控Bézier曲线的改进

目的 在用Bézier曲线表示复杂形状时,相邻曲线的控制顶点间必须满足一定的光滑性条件。一般情况下,对光滑度的要求越高,条件越复杂。通过改进文献中的“可调控Bézier曲线”,以构造具有多种优点的自动光滑分段组合曲线。方法 首先给出了两条位置连续的曲线G^l连续的一个充分条件,进而证明了“可调控Bézier曲线”在普通Bézier曲线的G^l光滑拼接条件下可达G^l(l为曲线中的参数)光滑拼接。然后对“可调控Bézier基”进行改进得到了一组新的基函数,利用该基函数按照Bézier曲线的定义方式构造了一种新曲线。分析了该曲线的光滑拼接条件,并根据该条件定义了一种分段组合曲线。结果 对于新曲线而言,只要前一条曲线的最后一条控制边与后一条曲线的第1条控制边重合,两条曲线便自动光滑连接,并且在连接点处的光滑度可以简单地通过改变参数的值来自由调整。由新曲线按照特殊方式构成的分段组合曲线具有类似于B样条曲线的自动光滑性和局部控制性。不同的是,组合曲线的各条曲线段可以由不同数量的控制顶点定义,选择合适的参数,可以使曲线在各个连接点处达到任何期望的光滑度。另外,改变一个控制顶点,至多只会影响两条曲线段的形状,改变一条曲线段中的参数,只会影响当前曲线段的形状,以及至多两个连接点处的光滑度。结论 本文给出了构造易于拼接的曲线的通用方法,极大简化了曲线的拼接条件。此基础上,提出的一种新的分段组合曲线定义方法,无需对控制顶点附加任何条件,所得曲线自动光滑,且其形状、光滑度可以或整体或局部地进行调整。本文方法具有一般性,为复杂曲线的设计创造了条件。     

平面参数曲线变形的仿射映射B样条方法

提出一种新的参数曲线变形方法:采用一种特殊的B样条展开式作为伸缩函数,构造了具有明确几何意义的变换矩阵,用它作用于待变形的曲线,可使曲线发生变形。此方法数学模型简单而变形效果良好。展开式的系数作为变形的控制参数,每个参数具有局部可控性,变形效果较丰富。可分别定量地控制变形的发生区间、变形区间界点处的连续性与光滑性、变形方向和变形幅度等。实验表明,该方法通过交互改变控制参数,可获得预期的、丰富的形状修改和变形效果,适用于几何造型、计算机动画、CAD等领域。     

Simultaneous and incremental feature-based multiresolution modeling with feature operations in part design

This paper proposes a new feature-based multiresolution modeling approach that can provide multiresolution representation of dynamically changing CAD models of intermediate design stage. Feature-based multiresolution modeling provides simplified shapes of parts of various levels of detail (LOD) by suppressing the detailed features according to a certain LOD criterion. Unlike previous research having mainly focused on the multiresolution representation of the final design model, our approach can carry out simultaneous and incremental multiresolution representation on the intermediate design models. To implement the system supporting this capability, we developed history-based selective Boolean operations, in which if the order of the Boolean operations for a part is altered, the regions affected by the operations are redefined according to the history of the Boolean operations so that the resultant shape may be the same as the original shape of the part. The system implemented using these operations guarantees a unique and valid shape for each intermediate LOD in the simultaneous multiresolution modeling environment. Since the system provides the designer immediately with various detail levels of the CAD model in any design stage, the design process is expected to be accelerated.     

准均匀B样条曲面的多分辨率表示及应用

在多分辨率曲线和曲面造型中，B样条小波已经得到了广泛应用。曲线和曲面的多分辨率造型成为一个研究热点。通过阐述准均匀B样条曲线曲面的基于小波分解的多分辨率表示的数学原理，给出了具体的曲线和曲面小波分解算法和实验结果，说明了准均匀B样条曲面多分辨表示的优点及其在工业上的应用。     

Reverse engineering applications for recovery of broken or worn parts and re-manufacturing: Three case studies

Damaged or broken parts are generally too expensive to replace, or are no longer available. RE can be defined as: ‘Systematic evaluation of a product with the purpose of replication. This involves design of a new part, copy of an existing part, recovery of a damaged or broken part, improvement of model precision and inspection of a numerical model. Advantages of the technique include immediate feedback, data reduction, direct generation of geometry and higher precision of the final product. This paper shows some possibilities of use and benefit from utilising the RE-methodologies and techniques in production process, especially in the case when exists parts without 3D-CAD support.This paper, which defines obtaining CAD data step by step from damaged three different parts to reproduce or make a new design for some recoveries, has not get any technical drawings. When these parts had been recovered, some problems occurred. These problems have been solved by referring to some practical approaches. Establishing continuity across curve and surface patches is an important concept in the free form surface modeling. The CAD models were recovered and reconstructed to consider parametric and geometric continuity. The iso-phote method was used for surface continuity analysis. Hence, in this work, not only occurring problems but also solving methods were explained. Firstly, CAD models are created from damaged and broken parts by data digitization method by using CMM and the process was explored in detailed. Later, CAD models that had been obtained earlier are transferred into CAM module of the software and G codes are taken by the NC post-processor, and finally, the parts are manufactured by means of CNC milling machine. Additionally, this paper presents a review of the part digitization techniques available for reverse engineering.     

Loop细分小波框架对图形传输与去噪的应用

基于细分小波的多分辨率分析是近年来三维图形处理的重要方法,该方法在图形的压缩,去噪,渐近显示和传输,多分辨率绘制和编辑等领域已有很多研究与应用。最近Maria Charina等人提出了一种新的基于细分小波紧框架的多分辨率分析理论,使得细分小波框架在三维图形处理领域的应用成为值得研究的新课题。在深入学习和研究这种全新的多分辨率分析理论的基础上,详细推导了Loop细分小波紧框架的分解和重构公式,应用这些公式实现了多分辨率曲面的构造并将其应用到三维网格图形的渐进传输和去噪中。通过与M.Bertram的双正交Loop细分小波算法的对比,表明基于Loop细分小波紧框架的多分辨率分析算法具有较好的渐进传输和去噪效果。由于通常的输入网格不具有细分连通性,而基于细分曲面的多分辨率分析算法要求它所处理的网格具有这种连通性,所以还特别提出了一种构造既能逼近输入网格又具有细分连通性的网格的简捷算法。     

A class of general quartic spline curves with shape parameters

With a support on four consecutive subintervals, a class of general quartic splines are presented for a non-uniform knot vector. The splines have C² continuity at simple knots and include the cubic non-uniform B-spline as a special case. Based on the given splines, piecewise quartic spline curves with three local shape parameters are given. The given spline curves can be C²∩G³ continuous by fixing some values of the curve?s parameters. Without solving a linear system, the spline curves can also be used to interpolate sets of points with C² continuity. The effects of varying the three shape parameters on the shape of the quartic spline curves are determined and illustrated.     

Computational procedure for optimum shape design based on chained Bezier surfaces parameterization

Optimum design introduces strong emphasis on compact geometry parameterization in order to reduce the dimensionality of the search space and consequently optimization run-time. This paper develops a decision support system for optimum shape which integrates geometric knowledge acquisition using 3D scanning and evolutionary shape re-engineering by applying genetic-algorithm based optimum search within a distributed computing workflow.A shape knowledge representation and compaction method is developed by creating 2D and 3D parameterizations based on adaptive chaining of piecewise Bezier curves and surfaces. Low-degree patches are used with adaptive subdivision of the target domain, thereby preserving locality. C¹ inter-segment continuity is accomplished by generating additional control points without increasing the number of design variables. The control points positions are redistributed and compressed towards the sharp edges contained in the data-set for better representation of areas with sharp change in slopes and curvatures. The optimal decomposition of the points cloud or target surface into patches is based on the requested modeling accuracy, which works as lossy geometric data-set compression. The proposed method has advantages in non-recursive evaluation, possibility of chaining patches of different degrees, options of prescribing fixed values at selected intermediate points while maintaining C¹ continuity, and uncoupled processing of individual patches.The developed procedure executes external application nodes using mutual communication via native data files and data mining. This adaptive interdisciplinary workflow integrates different algorithms and programs (3D shape acquisition, representation of geometry with data-set compaction using parametric surfaces, geometric modeling, distributed evolutionary optimization) such that optimized shape solutions are synthesized. 2D and 3D test cases encompassing holes and sharp edges are provided to prove the capacity and respective performance of the developed parameterizations, and the resulting optimized shapes for different load cases demonstrate the functionality of the overall distributed workflow.     

Wavelet-Based Multiresolution with

Multiresolution methods are a common technique used for dealing with large-scale data and representing it at multiple levels of detail. We present a multiresolution hierarchy construction based on subdivision, which has all the advantages of a regular data organization scheme while reducing the drawback of coarse granularity. The -subdivision scheme only doubles the number of vertices in each subdivision step regardless of dimension n. We describe the construction of 2D, 3D, and 4D hierarchies representing surfaces, volume data, and time-varying volume data, respectively. The 4D approach supports spatial and temporal scalability. For high-quality data approximation on each level of detail, we use downsampling filters based on n-variate B-spline wavelets. We present a B-spline wavelet lifting scheme for -subdivision steps to obtain small or narrow filters. Narrow filters support adaptive refinement and out-of-core data exploration techniques.