Geometric Modelling

In many areas of industry, it is desirable to create geometric models of existing objects for which no such model is available. This paper reviews the process of reverse engineering of shapes. After identifying the purpose of reverse engineering and the main application areas, the most important algorithmic steps are outlined and various reconstruction strategies are presented. Pros and cons of various data acquisition techniques are described with related problems of boundary representation model construction.

Current geometric and solid modeling systems use semi-algebraic sets for de ning the boundaries of solid objects, curves and surfaces, geometric constraints with mating relationship in a mechanical assembly, physical contacts between objects, collision detection. It turns out that performing many of the geometric operations on the solid boundaries or interacting with geometric constraints is reduced to nding common solutions of the polynomial equations. Current algorithms in the literature based on symbolic, numeric and geometric methods su er from robustness, accuracy or e ciency problems or are limited to a class of problems only.

An original method has been developed to model geology using the location of the geological interfaces and orientation data from structural field. Both types of data are cokriged to interpolate a continuous 3D potential-field scalar function describing the geometry of the geology. Geology contact locations set the position of reference isovalues while orientation data are the gradients of the scalar function. Geometry of geological bodies is achieved by discretising reference isovalues. Faults are modelled using the same method by inserting discontinuities in the potential field. Potential fields can be combined to model realistic, complex geometry: scalar functions representing separate geological series are merged automatically using geological rules to enable fast computation and easy update of interpretation. The methodology has been applied to a wide range of geological contexts including orogenic domains, basins, intrusive and extrusive environments.

BSP trees have been shown to provide an effective representation of polyhedra through the use of spatial subdivision, and are an alternative to the topologically based b-reps. While bsp tree algorithms are known for a number of important operations, such as rendering, no previous work on bsp trees has provided the capability of performing boolean set operations between two objects represented by bsp trees, i.e. there has been no closed boolean algebra when using bsp trees. This paper presents the algorithms required to perform such operations. In doing so, a distinction is made between the semantics of polyhedra and the more fundamental mechanism of spatial partitioning. Given a partitioning of a space, a particular semantics is induced on the space by associating attributes required by the desired semantics with the cells of the partitioning. So, for example, polyhedra are obtained simply by associating a boolean attribute with each cell. Set operations on polyhedra are then constructed on top of the operation of merging spatial partitionings. We present then the algorithm for merging two bsp trees independent of any attributes/semantics, and then follow this by the additional algorithmic considerations needed to provide set operations on polyhedra. The result is a simple and numerically robust algorithm for set operations.

Generating realistic 3D human face models and facial animations has been a persistent challenge in computer graphics. We have developed a system that constructs textured 3D face models from videos with minimal user interaction. Our system takes images and video sequences of a face with an ordinary video camera. After five manual clicks on two images to tell the system where the eye corners, nose top and mouth corners are, the system automatically generates a realistic looking 3D human head model and the constructed model can be animated immediately. A user, with a PC and an ordinary camera, can use our system to generate his/her face model in a few minutes.

Industrial Geometry aims at unifying existing and developing new methods and algorithms for a variety of application areas with a strong geometric component. These include CAD, CAM, Geometric Modelling, Robotics, Computer Vision and Image Processing, Computer Graphics and Scientific Visualization. In this paper, Industrial Geometry is illustrated via the fruitful interplay of the areas indicated above in the context of novel solutions of CAD related, geometric optimization problems involving distance functions: approximation with general B-spline curves and surfaces or with subdivision surfaces, approximation with special surfaces for applications in architecture or manufacturing, approximate conversion from implicit to parametric (NURBS) representation, and registration problems for industrial inspection and 3D model generation from measurement data. Moreover, we describe a 'feature sensitive' metric on surfaces, whose definition relies on the concept of an image manifold, introduced into Computer Vision and Image Processing by Kimmel, Malladi and Sochen. This metric is sensitive to features such as smoothed edges, which are characterized by a significant deviation of the two principal curvatures. We illustrate its applications at hand of feature sensitive curve design on surfaces and local neighborhood definition and region growing as an aid in the segmentation process for reverse engineering of geometric objects.

Parameterization of 3D mesh data is important for many graphics and mesh processing applications, in particular for texture mapping, remeshing and morphing. Closed, manifold, genus-0 meshes are topologically equivalent to a sphere, hence this is the natural parameter domain for them. Parameterizing a 3D triangle mesh onto the 3D sphere means assigning a 3D position on the unit sphere to each of the mesh vertices, such that the spherical triangles induced by the mesh connectivity do not overlap. This is called a spherical triangulation. In this paper we formulate a set of necessary and sufficient conditions on the spherical angles of the spherical triangles for them to form a spherical triangulation. We formulate and solve an optimization procedure to produce spherical triangulations which reflect the geometric properties of a given 3D mesh in various ways.

Self-organized TiO 2 nanotubes with packed, vertically aligned morphology and different lateral characteristics were grown on Ti metal substrates by controlled electrochemical anodization in phosphate/HF and ethylene glycol/HF electrolytes. The wetting, photo-induced superhydrophilicity, and photocatalytic activity of the nanotubular materials were investigated under ultraviolet irradiation. The photoactivity of the TiO 2 nanotube arrays was analysed in terms of their morphological characteristics that were determined by means of scanning electron microscopy and atomic force microscopy in conjunction with geometrical modelling. The wetting and the UV-induced superhydrophilicity could be accordingly modelled by the Cassie-Baxter mode arising from the large scale roughness of the nanotubular arrays in combination with the Wenzel mode due to the small scale roughness induced by ridges at the outer tube surface. The photocatalytic activity of the TiO 2 nanotube arrays was further found to correlate quantitatively with the variation of the geometric roughness factor, verifying the strong impact of morphology on the photo-induced properties of the vertically oriented TiO 2 tubular architecture.

This article reviews the prevailing geometric modelling techniques, based on Non-Uniform Rational B-Splines (NURBS). Emphasis is placed on the most important properties of NURBS surfaces and the available techniques for modelling real natural or artificial objects given a cloud of three-dimensional data points on their surface, possibly taken from a scanning device.

The presence of holes in a triangle mesh is classically ascribed to the deficiencies of the point cloud acquired from a physical object to be reverse engineered. This lack of information results from both the scanning process and the object complexity. The consequences are simply not acceptable in many application domains (e.g. visualisation, finite element analysis or STL prototyping). This paper addresses the way these holes can be filled in while minimizing the curvature variation between the surrounding and inserted meshes. The curvature variation is simulated by the variation between external forces applied to the nodes of a linear mechanical model coupled to the meshes. The functional to be minimized is quadratic and a set of geometric constraints can be added to further shape the inserted mesh. In addition, a complete cleaning toolbox is proposed to remove degenerated and badly oriented triangles resulting from the scanning process.

The proximity and topology among particles are often the most important factor for understanding the spatial structure of particles. Reasoning the morphological structure of molecules and reconstructing a surface from a point set are examples where proximity among particles is important. Traditionally, the Voronoi diagram of points, the power diagram, the Delaunay triangulation, and the regular triangulation, etc. have been used for understanding proximity among particles. In this paper, we present the theory of the β-shape and the β-complex and the corresponding algorithms for reasoning proximity among a set of spherical particles, both using the quasi-triangulation which is the dual of the Voronoi diagram of spheres. Given the Voronoi diagram of spheres, we first transform the Voronoi diagram to the quasi-triangulation.

Discrete fracture network simulations are computationally intensive and usually time-consuming to construct and configure. This paper presents a case study with techniques for building a 3D finite element model of an inhomogeneous fracture network for modelling flow and tracer transport, combining deterministic and stochastic information on fracture aperture distributions. The complex intersected fractures represent a challenge for geometrical model design, mesh quality requirements and property allocations. For the integrated and holistic modelling approach, including the application of numerical and analytical simulation techniques, new object-oriented concepts in software engineering are implemented to ensure a resourceful and practicable software environment.

Much work on special elements that simplify geometrical modelling of structures containing holes, cracks and/ or inclusions has been reported extensively in the literature. This paper presents a hybrid Trefftz element containing elliptic hole formulated using Hellinger-Reissner principle by employing trial functions based on the mapping technique and the Cauchy integral method. The element presented in this paper could be regarded as an improved formulation over Piltner [4] element because the chosen trail functions in this paper have provided relatively more stable solutions. The use of the element with other ordinary displacement based finite elements has also yielded very accurate solutions even when very coarse meshes relative to the size of the elliptic hole have been used.

In this paper, we present an FL-system, an extension of an L-system that allows us to generate any kind of object hierarchy and mesh on the fly. This has been made possible thanks to a modification of the classical L-system rewriting mechanism that produces a string of symbols interpreted afterwards. In our system, terminal symbols are not characters, but functions that can be executed at any step of the rewriting process. Thanks to this extension, our system allows the instantiation of generic objects during the course of the rewriting process as well as their initialization. Therefore, we are able to simulate all of the existing solutions proposed by classical Lsystems, but we are also able to generate VRML97 scene graphs and geometry on the fly, since VRML97 nodes are handled as generic objects.

Computing boolean operations between general planar polygons is one of the fundamental problems in geometric and solid modeling. In this work we present a new algorithm to calculate intersection, union and di!erence, valid for general planar polygons (manifold and non-manifold, with and without holes), based on a formal representation system. This formal model is based on the concept of simplicial chain, developed by Feito and Rivero (Computers & Graphics 22(5) (1998)). Using algebraic operations between simplicial chains we can obtain any general polygon and the Boolean operations between them. The fact of that our algorithm is based on simplicial chains and their operations, reduces the study of special cases, and allows us to develop a robust and e$cient algorithm to calculate the intersection between general polygons.

In our paper, we attempt to investigate methods of determining jersey porosity which is this fabric's principal physical characteristic. In fact, end use, liquid absorbency, thermal comfort and resistance are closely related to pore size and distribution. So it is important to study porosity, in order to classify and determine the right use of jersey knitted structure. Many methods are used to estimate porosity, but most concern air permeability, image processing and geometry modelling. The first mentioned is used for the stretched structure, the second is valid for fabrics with high porosity levels, and the last mentioned is used to confirm any structure's conformation. The aims of this study are twofold; firstly, to recognise the most suitable and easiest method of estimating the fabric's porosity, and secondly to study the influence on porosity of various knitting parameters of jersey structure such as yarn number and count, fabric thickness, loop length, and stitch density.

Fault-related folds in fold-and-thrust belts and accretionary prisms are commonly simulated by three end-member classes of geometric models: décollement folding, fault-propagation folding, and fault-bend folding. A given model produces a fold shape that depends on the amount of shortening, the fault shape and, eventually, the fault propagation history. The great variety of geometric solutions implies that the final fold shape is adequate to distinguish among different folding mechanisms only in a limited suite of cases. Additional information to better constrain fold kinematics is provided by the analysis of growth strata patterns, which passively register the progressive evolution of deformation in the substratum. Availability of growth strata pattern templates associated with different geometric and kinematic solutions provides a useful tool for inferring the dominant kinematic pathways of natural thrust-related structures. In this paper, we present growth strata patterns associated with double-edge fault-propagation folding. The influence of the sedimentation rate, the fault shape, and the fault S/P ratio (i.e. slip versus ramp propagation rate) on growth strata architectures are investigated. Geometrical modelling and application to natural examples indicate that double-edge fault-propagation anticlines are characterised by growth triangles on both limbs. Their position and geometry relate to the ramp shape and propagation history, and on the sedimentation rate.

Geometrical modelling and field analysis of chevron folds suggest that these structures are a result of the combination of several kinematical mechanisms, whose magnitude and order of application vary within certain limits. A possible mechanism operating early in the fold growth is homogeneous layer shortening, whose contribution is restricted by the high competence contrast that the multilayers developing chevron folds usually exhibit. In the early stages of folding, when the curvature is small, equiareal tangential longitudinal strain (ETLS) is an essential mechanism, since the operation of parallel tangential longitudinal strain (PTLS) or flexural flow (FF) would give rise respectively to area changes or strain in the limbs of the final fold that are too high to be geologically realistic. After folding by ETLS, probable mechanisms are PTLS and FF, which can operate in this order or simultaneously. In general, FF is necessary at the last stage of buckling, although the increment of folding due to this mechanism can be very small. High values of slip between layers and area change produced in the later stages of chevron folding can bring an end to buckling, probably at an interlimb angle value of 60e70 , and induce the onset of homogeneous strain (HS). This strain is not coaxial in many cases, with simple shear playing an important role, and gives rise to asymmetrical folds.

In [1] Optimal Control methods over re-parametrization for curve and surface design were introduced. The advantage of Optimal Control over Global Minimization such as in is that it can handle both approximation and interpolation. Moreover a cost function is introduced to implement a design objective (shortest curve, smoothest one etc...). The present work introduces the Optimal Control over the knot vectors of non-uniform B-Splines.

In the last ve y ears, there has been numerous applications of wavelets and m ultiresolution analysis in many elds of computer graphics as di erent as geometric modelling, volume visualization or illumination modelling. Classical multiresolution analysis is based on the k n o wledge of a nested set of functional spaces in which t he s u ccessive a p proximations of a given function converge to t hat f u nction, and can be e ciently computed. This paper rst proposes a theoretical framework which e n ables multiresolution analysis even if the f u nctional spaces are not nested, as long a s t hey still have t he property that t he s u ccessive a p proximations converge to t he g i v en function. Based on this concept we n ally introduce a new multiresolution analysis with exact reconstruction for large data sets de ned on uniform grids. We construct a one-parameter family of multiresolution analyses which is a blending o f H a a r and linear multiresolution.

Existing 3-dimensional acoustic target tracking methods that use wired/wireless networked sensor nodes to track targets based on four sensing coverage do not always compute the feasible spatio-temporal information of target objects. To investigate this discrepancy in a formal setting, we propose a geometric model of the target tracking problem alongside its equivalent geometric dual model that is easier to solve. We then study and prove some properties of dual model by exploiting its relationship with algebra. Based on these properties, we propose a four coverage axis line method based on four sensing coverage and prove that four sensing coverage always yields two dual correct answers; usually one of them is infeasible. By showing that the feasible answer can be only sometimes identified by using a simple time test method such as the one proposed by ourselves, we prove that four sensing coverage fails to always yield the feasible spatiotemporal information of a target object. We further prove that five sensing coverage always gives the feasible position of a target object under certain conditions that are discussed in this paper. We propose three extensions to four coverage axis line method, namely, five coverage extent point method, five coverage extended axis lines method, and five coverage redundant axis lines method. Computation and time complexities of all four proposed methods are equal in the worst cases as well as on average being equal to Θ 1 each.

Geometric Modellers based on a Boundary Representation scheme are well suited for display operations, but boolean operations require algorithms with quadratic complexity. Using a class of extended Octtrees which is presented here, boolean operations become linear, while exact recomputation of the boundary model in the case of polyhedral objects is possible, and memory requirements are less than in classical octtree encodings. Some bounds on the memory savings are given and discussed. The DMI system, a Geometric Modeller based on a hybrid model Boundary -Extended Octtrees, is presented, discussing the main operations and the interface between both models. Some modelling examples are shown. Finally, a generalization of the extended Octtree encoding which allows the exact representation of objects limited by sculptured surfaces is presented.

Computer aided geometric design is an area where the betterment of surface generation techniques is an everlasting demand since faster and more accurate geometric models are required. The traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling and computer graphics since they offer a number of features from which these areas can benefit. PDEs have been utilised under different approaches that vary from the development of surface generation techniques to applications addressing concrete problems such as noise reduction. This work summarises the uses given to PDE surfaces as a surface generation technique together with some other applications to computer graphics.

Increasing complexity in structural shapes, the need for more detailed and precise numerical simulations, and shorter product development times request a tightly connected interaction within the design software environment since both fields, geometric design and structural analysis, rely on the results of each other. Therefore, this paper proposes an integrated design process, which is predicated on a NURBS-based CAD environment as well as on the NURBS-based isogeometric analysis, and substantiates in a mechanical point of view. The presented investigations are concentrated on thin-walled structures. In this paper several issues, e.g. continuity conditions associated with the applied finite element type, arising from the increased challenges for the geometric modelling within an integrated design process are identified and solution strategies are discussed. Furthermore, general analysis-aware modelling guidelines are derived. The paper concludes by exemplifying the complete integrated structural design process for a structural part. Keywords Integrated structural design process • Integration of CAD and CAE • Isogeometric analysis • NURBS • Analysis-suitable geometric modelling

Experimental studies of fatigue crack growth in aluminium alloys have shown that, at the low-to-mid stress intensity factor range, there is often a log-linear relationship between the crack length and the fatigue life. These observations have led to the development of the generalised Frost-Dugdale crack growth law, which allowed the accurate prediction of fatigue crack growth from Region I. For this research paper the 'generalised Frost-Dugdale' law was used to perform an optimisation study of 7050-T7451 Aluminium structures. The structural optimisation procedure proposed integrates geometrical modelling, structural analysis and optimization into one complete and automated computer-aided design process. The results from the structural optimisation study compared the 'generalised Frost-Dugdale' law and the traditional Paris law. Gradient-less, gradient-based optimisation algorithm and an enumeration scheme were considered in this investigation. The enumeration scheme takes advantage of a cluster computer architecture which enables a visualisation of the solution space allowing verification and validation of the optimisation algorithm. The results indicated that the optimal geometrical shape and predicted fatigue life depended on the crack sizes, structural geometry, boundary conditions and fatigue crack growth law. As a result, this procedure illustrates that for the design of light weight structures, a fatigue based optimisation used in conjunction with visualisation of the solution space may provide a viable design methodology. The importance of non-destructive inspection (NDI) and its role in determining optimal structural geometries is also revealed. Furthermore, the possibility of the application of the generalised Frost-Dugdale model in design optimisation has been demonstrated. This procedure has the potential to be applied to structures with complex structural configurations taking into account crack propagation in Region I.

In this paper we present some former results about properties of digital circles defined by neighbourhood sequences in the triangular grid. Das and Chatterji [Inform. Sci. 50 (1990) 123] analyzed the geometric behaviour of twodimensional periodic neighbourhood sequences. We use a more general definition of neighbourhood sequences, which does not require periodicity [Publ. Math. Debrecen 60 (2002) 405]. We study the development of wave-fronts and grow digital circles from a triangle with general neighbourhood sequences in triangular grid. We present the possible types of polygons, and characterize them by the initial part of the neighbourhood sequences. The symmetry and the convexity analysis of the digital circles is also presented. Besides those who are interested in the underlying theory there may be readers from the pattern recognition or the image processing communities or even the geometric modelling field who could find some of the consequences of the paper of interest.

Unlike many other works, where authors are usually focused on one or two quality criteria, the current manuscript, which is a generalization of the article [35] published in Russian, offers a multi-criteria approach to the assessment of the shape quality of curves that constitute component parts of the surfaces used for the computer modelling of object shapes in various types of design. Based on the analysis of point particle motion along a curved path, requirements for the quality of functional curves are proposed: a high order of smoothness, a minimum number of curvature extrema, minimization of the maximum value of curvature and its variation rate, minimization of the potential energy of the curve, and aesthetic analysis from the standpoint of the laws of technical aesthetics. The authors do not set themselves the task of giving a simple and precise mathematical definition of such curves. On the contrary, this category can include various curves that meet certain quality criteria, the refinement and addition of which is possible in the near future. Engineering practice shows that quality criteria can change over time, which does not diminish the need to develop multi-criteria methods for assessing the quality of geometric shapes. Technical issues faced during edge rounding in 3D models that affect the quality of industrial design product shape have been reviewed as an example of the imperfection of existing CAD systems.

The estimation of body segment properties is important in the biomechanical analysis of movement. Current subject-specific estimation methods however can be expensive and time-consuming, while other methods do not adequately take into account individual or group variability. We describe a simple procedure for estimating subject-specific geometric properties, independent of joint centres. The method requires only a small number of anthropometric measurements and digital images of the segment or subject, a 3-dimensional modeller program and simple mathematical calculations to estimate segment volumes and centroids. Assuming that the segment is of uniform density, it's mass and moment of inertia can also be derived. Future work should include generating segment density profiles for particular populations, to increase the accuracy of the method, and comparing the accuracy of the results obtained with those produced by other techniques.

Digital elevation model (DEM) extracted from IKONOS along-track stereo images with photogrammetric method is evaluated. As few as 12 GCPs are enough for the stereo photogrammetric bundle adjustment, which also filters the errors of the input data. With an area-based image matching users may produce high resolution DEMs with LE68 errors of 1 m to 4 m depending on the land covers. The best results (1.1 m-2.6 m) are obtained in bare soils, lakes, residential areas and sparse forests. The surface elevation of some of the areas (residential/ forests) did not affect too much the errors because the 1-2-storey houses in residential areas are sparse or because the images were acquired when there is no leave in the deciduous forests. An error evaluation as a function of the slope azimuths shows that the DEM error in sun-facing slopes is 1-m smaller than the DEM error in slopes away from the sun. 5-10 m contour lines could thus be derived with the highest topographic standard.

In this paper, a family of adaptive mesh generation schemes specially designed for finite element modelling of structural hollow section (SHS) tubular joint is presented. This family of adaptive mesh generation schemes is implemented based on a series of realistic and consistent geometrical models which are founded on measurements obtained from real structures. The underlying geometrical models provide the definitions of different levels of geometrical details and special features that could appear during the entire life cycle of the structure. The adaptive mesh generation schemes accompanying the geometrical models are capable of discretizing the SHS tubular joints into different forms of finite element meshes including pure surface meshes, hybrid meshes with surface and solid elements, and full 3D solid element meshes with or without weld and crack details. As a result, a hierarchical adaptive modelling procedure could be developed to assess the performance of the structures for their whole life cycle from quasi-static failure strength analysis to long-term fatigue and fracture behaviours under cyclic loadings. In addition, all the mesh generators in this family are adaptive mesh generators such that the discretization error of the corresponding FE models could be effectively controlled by combining them with appropriate adaptive refinement schemes.

Several fields of mathematics are relevant to computer aided design and other software systems involving solid object geometry, topology, differential and algebraic geometry being particularly important. This paper discusses some of this mathematics in order to provide a theoretical foundation for geometric modelling kernels that support non-manifold objects with an internal cellular structure and subsets of different dimensions.

In this study we analyze the chromatin state of human pluripotent stem cells by geometric and computational modelling of fibre conformation. The model takes into account local structure of chromatin organized into euchromatin, permissive for gene activation, and heterochromatin, transcriptionally silenced. Euchromatin was modelled using linear DNA while heterochromatin by means of a solenoid structure in which DNA winds onto six nucleosome spools per turn. Two geometric models are presented and are compared in terms of geometric quantities. The models are tested using in vivo data generated from chromatin human immunoprecipitation from embryonic stem cells. This study provides insight for identifying the relationships between chromosome geometry and epigenomic processes associated with chromatin remodeling, cellular reprograming and maintenance of cellular pluripotency.

This paper presents a study of moisture retention and isothermal transfer in a cement and lime mortar. The extended range of pore sizes and the intrinsic heterogeneity of the medium do not allow the use of a single-scale percolation network for numerically describing the topology and the physical processes related to fluid retention and transfer. Three different scales are used for the numerical description of the cement and lime mortar studied in this paper. Electron scanning pictures are used for the geometrical modelling of the material in the three different scales. The heterogeneity is associ,~ed with the fissures network, represented in the first scale, where the medium was conceived as a series of cubic blocks separated by straight channels. The second and third scale are used to represent the pores in the cement and lime paste. A 3D simple cubic percolation site network is used in the third scale to model imbibition and drainage and to predict the hydraulic conductivity in terms of the moisture content. In the second scale, pores are represented as embedded bodies in a continuous medium and the hydraulic conductivity is calculated using Maxwell De Vries theory for composite media. At the first scale level, the isothermal mass diffusivity D~ is modelled by studying an invasion process of liquid water into the fissures network, simulating an actual imbibition experiment, from one extremity of a sample column. Finally, the results of the simulation are compared with experimentally obtained values of Do.

Computer-aided surgical simulation is a topic of increasingly extensive research. Computer graphics, geometric modeling and finite-element analysis all play major roles in these simulations. Furthermore, real-time response, interactivity and accuracy are crucial components in any such simulation system. A major effort has been invested in recent years to find ways to improve the performance, accuracy and realism of existing systems.

The Reverse Engineering process consists of a succession of operations that aim at creating a digital representation of a physical model. The reconstructed geometric model is often a triangle mesh built from a point cloud acquired with a scanner. Depending on both the object complexity and the scanning process, some areas of the object outer surface may never be accessible, thus inducing some deficiencies in the point cloud and, as a consequence, some holes in the resulting mesh. This is simply not acceptable in an integrated design process where the geometric models are often shared between the various applications (e.g. design, simulation, manufacturing). In this paper, we propose a complete toolbox to fill in these undesirable holes. The hole contour is first cleaned to remove badly-shaped triangles that are due to the scanner noise. A topological grid is then inserted and deformed to satisfy blending conditions with the surrounding mesh. In our approach, the shape of the inserted mesh results from the minimization of a quadratic function based on a linear mechanical model that is used to approximate the curvature variation between the inner and surrounding meshes. Additional geometric constraints can also be specified to further shape the inserted mesh. The proposed approach is illustrated with some examples coming from our prototype software.

This paper describes a new method for interactive segmentation that is based on cross-sectional design and 3D modelling. The method represents a 3D model by a set of connected contours that are planar and orthogonal. Planar contours overlayed on image data are easily manipulated and linked contours reduce the amount of user interaction. This method solves the contour-to-contour correspondence problem and can capture extrema of objects in a more exible way than manual segmentation of a stack of 2D images. The resulting 3D model is guaranteed to be free of geometric and topological errors. We show that manual segmentation using connected orthogonal contours has great advantages over conventional manual segmentation. Furthermore, the method provides e ective feedback and control for creating an initial model for, and control and steering of, (semi-)automatic segmentation methods. ?

The next generation manufacturing technologies will draw on new developments in geometric modelling. Based on a comprehensive analysis of the desiderata of next generation geometric modellers, we present a critical review of the major modelling paradigms, namely, CSG, B-Rep, non-manifold, and voxel models. We present arguments to support the view that voxel-based modellers have attributes that make it the representation scheme of choice in meeting the emerging requirements of geometric modelling.