Advances in Geometric Modeling and Processing

We propose an automatic PolyCube-Maps construction scheme. Firstly, input mesh is decomposed into a set of feature regions, and further split into patches. Then, each region is approximated by a simple basic polycube primitive with each patch mapped to a rectangular sub-surface of the basic polycube primitive which can be parameterized independently. After that, an iterative procedure is performed to improve the parameterization quality globally. By these steps, we can obtain the polycubic parameterization result efficiently.

We present a piecewise bi-cubic parametric G

1

spline surface interpolating the vertices of any irregular quad mesh of arbitrary topological type. While tensor product surfaces need a chess boarder parameterization they are not well suited to model surfaces of arbitrary topology without introducing singularities. Our spline surface consists of tensor product patches, but they can be assembled with G

1

-continuity to model any non-tensor-product configuration. At common patch vertices an arbitrary number of patches can meet. The parametric domain is built by 4-splitting one unit square for each input quadrangular face. This key idea of our method enables to define a very low degree surface, that interpolates the input vertices and results from an explicit and local procedure : no global linear system has to be solved.

Given a control mesh of a Loop subdivision surface, by pushing the control vertices to their limit positions, a

limit mesh

of the Loop surface is obtained. Compared with the control mesh, the limit mesh is a tighter linear approximation in general, which inscribes the limit surface. We derive an upper bound on the distance between a Loop subdivision surface patch and its limit triangle in terms of the maximum norm of the mixed second differences of the initial control vertices and a constant that depends only on the valence of the patch’s extraordinary vertex. A subdivision depth estimation formula for the limit mesh approximation is also proposed.

The topology of polygonal meshes has a large impact on the performance of various geometric processing algorithms, such as rendering and collision detection algorithms. Several approaches for simplifying topology have been discussed in the literature. These methods operate locally on models, which makes their effect on topology hard to predict and analyze. Most existing methods also tend to exhibit various disturbing artifacts, such as shrinking of the input and splitting of its components. We propose a novel top-down method for topology simplification that avoids the problems common in existing methods. The method starts with a simple, genus-zero mesh that bounds the input and gradually introduces topological features by a series of carving operations. Through this process a multiresolution stream of meshes is created with increasing topologic level of detail. Following the proposed approach, we present a practical carving algorithm that is based on the Constrained Delaunay Tetrahedralization (CDT). The algorithm pretetrahedralizes the complement of the input with respect to its convex hull and then eliminates tetrahedra in a prioritized manner. We present quality results for two families of meshes that are difficult to simplify by all existing methods known to us - topologically complex and highly clustered meshes.

This paper gives a method of quantifying small visual differences between 3D mesh models with conforming topology, based on the theory of strain fields. Our experiments show that our difference estimates are well correlated with human perception of differences. This work has applications in the evaluation of 3D mesh watermarking, 3D mesh compression reconstruction, and 3D mesh filtering.

This paper presents a novel approach which continuously detects the first collision between two curved-edge polygons moving under rational motions. Edges of the two polygons in this paper are planar curves, represented as conic splines, i.e. elliptic or parabolic sections. The curved-edge polygons are not confined to be convex and conic sections are only required to be

GC

0

continuous. Motions of the polygons are modeled by interpolating between control points along motion trajectories. Our algorithm returns the first collision moment and collision position if there is a collision between the two moving polygons and returns no-collision otherwise. Collision condition of the two polygons moving under rational motions is represented as an univariate polynomial of time t. Bernstein form is used to improve the accuracy of solving the high degree polynomial. We also use bounding circles to improve the efficiency of our approach and compare our method with the PIVOT2D method and prove ours to be more accurate and faster.

We present a method for computing the domain, where a control point is free to move so that the corresponding spatial curve is regular and of constant sign of torsion along a subinterval of its parametric domain of definition. The approach encompasses all curve representations that adopt the

control-point paradigm

and is illustrated for a spatial Bézier curve.

A finite element simulation framework for cutting and fracturing model without remeshing is presented. The main idea of proposed method is adding a discontinuous function for the standard approximation to account for the crack. A feasible technique is adopted for dealing with multiple cracks and intersecting cracks. Several involved problems including extended freedoms of finite element nodes as well as mass matrix calculation are discussed. The presented approach is easy to simulate object deformation while changing topology. Moreover, previous methods developed in standard finite element framework, such as the stiffness warping method, can be extended and utilized.

We propose a method to identify planar regions in volume data using a specialized version of the discrete Radon transform operating on a structured or unstructured grid. The algorithm uses an efficient discretization scheme for the parameter space to obtain a running time of

$\mathcal O(N (T\log T))$

, where

T

is the number of cells and

N

is the number of plane normals in the discretized parameter space.

We apply our algorithm in an industrial setting and perform experiments with real-world data generated by topology optimization algorithms, where the planar regions represent portions of a mechanical part that can be built using steel plate.

The

directional contact range

of two convex polyhedra is the range of positions that one of the polyhedron may locate along a given straight line so that the two polyhedra are in collision. Using the contact range, one can quickly classify the positions along a line for a polyhedron as “safe” for free of collision with another polyhedron, or “unsafe” for the otherwise. This kind of contact detection between two objects is important in CAD, computer graphics and robotics applications. In this paper we propose a robust and efficient computation scheme to determine the directional contact range of two polyhedra. We consider the problem in its dual equivalence by studying the Minkowski difference of the two polyhedra under a duality transformation. The algorithm requires the construction of only a subset of the faces of the Minkowski difference, and resolves the directional range efficiently. It also computes the contact configurations when the boundaries of the polyhedra are in contact.

We perform collision detection between static rigid objects using a bounding volume hierarchy which consists of an oriented bounding box (OBB) tree enhanced with bounding spheres. This approach combines the compactness of OBBs and the simplicity of spheres. The majority of distant objects are separated using the simpler sphere tests. The remaining objects are in close proximity, where some separation axes are significantly more effective than others. We select 5 from among the 15 potential separating axes for OBBs. Experimental results show that our algorithm achieved favorable speed up with respect to the existing OBB algorithms.

The paper proposes a fast fairing algorithm for curves and surfaces. It first defines a base algorithm for fairing curves, which is then extended to the surface case, where the isocurves of the surface are faired. The curve fairing process involves the discrete integration of a pseudo-arc-length parameterization of B-spline curves, with a blending and fitting phase concluding the algorithm. In the core of the fairing method, there is a fairness measure introduced in an earlier paper of the authors. This measure is based on the deviation from an ideal or target curvature. A target curvature is a series of smooth curvature values, generated from the original curve or surface. This curve and surface fairing technique is local and semi-automatic, but the user can also designate the region to be faired. The results are illustrated by a few examples on real-life models.

A variational formulation of a general form fourth order geometric partial differential equation is derived, and based on which a mixed finite element method is developed. Several surface modeling problems, including surface blending, hole filling and surface mesh refinement with the

G

1

continuity, are taken into account. The used geometric partial differential equation is universal, containing several well-known geometric partial differential equations as its special cases. The proposed method is general which can be used to construct surfaces for geometric design as well as simulate the behaviors of various geometric PDEs. Experimental results show that it is simple, efficient and gives very desirable results.

We study the problem of convergence of geodesics on PL-surfaces and in particular on subdivision surfaces. More precisely, if a sequence

of PL-surfaces converges in distance and in normals to a smooth surface

S

and if

C

n

is a geodesic of

T

n

(

i.e.

it is locally a shortest path) such that

converges to a curve

C

, we wonder if

C

is a geodesic of

S

. Hildebrandt et al. [11] have already shown that if

C

n

is a shortest path, then

C

is a shortest path. In this paper, we provide a counter example showing that this result is false for geodesics. We give a result of convergence for geodesics with additional assumptions concerning the rate of convergence of the normals and of the lengths of the edges. Finally, we apply this result to different subdivisions surfaces (such as Catmull-Clark) assuming that geodesics avoid extraordinary vertices.

We present algorithms to derive the precise Hausdorff distance and/or the minimal distance between two freeform shapes, either curves or surfaces, in

or

. The events at which the Hausdorff/minimal distance can occur are identified and means to efficiently compute these events are presented. Examples are also shown and the extension to arbitrary dimensions is briefly discussed.

Interpolation of a sequence of points by spline curves generally requires the solution of a large system of equations. In this paper we provide a method which requires only local computation instead of a global system of equations and works for a large class of curves. This is a generalization of a method which previously developed for B-spline, NURBS and trigonometric CB-spline curves. Moreover, instead of numerical shape parameters we provide intuitive, user-friendly, control point based modification of the interpolating curve and the possibility of optimization as well.

We introduce a new Lepp-Delaunay algorithm for quality triangulation. For every bad triangle t with smallest angle less than a threshold angle

θ

, a Lepp-search is used to find an associated convex terminal quadrilateral formed by the union of two terminal triangles which share a local longest edge (terminal edge) in the mesh. The centroid of this terminal quad is computed and Delaunay inserted in the mesh. The algorithm improves the behavior of a previous Lepp-Delaunay terminal edge midpoint algorithm. The centroid method computes significantly smaller triangulation than the terminal edge midpoint variant, produces globally better triangulations, and terminates for higher threshold angle

θ

(up to 36°). We present geometrical results which explain the better performance of the centroid method. Also the centroid method behaves better than the off-center algorithm for

θ

bigger than 25°.

Bernstein polynomials are a classical tool in Computer Aided Design to create smooth maps with a high degree of local control. They are used for the construction of Bézier surfaces, free-form deformations, and many other applications. However, classical Bernstein polynomials are only defined for simplices and parallelepipeds. These can in general not directly capture the shape of arbitrary objects. Instead, a tessellation of the desired domain has to be done first.

We construct smooth maps on arbitrary sets of polytopes such that the restriction to each of the polytopes is a Bernstein polynomial in mean value coordinates (or any other generalized barycentric coordinates). In particular, we show how smooth transitions between different domain polytopes can be ensured.

Extended from the 2D silhouette-parsing short-cut rule [25], a 3D short-cut rule, which states “ as long as a cutting path mainly crosses local skeleton and lies in concave regions, the shorter path is (other things being equal) the better ” , is defined in the paper. Guided by the 3D short-cut rule, we propose a hierarchical model decomposition paradigm, which integrates the advantages of the skeleton-driven and minima-rule-based meaningful segmentation. Our method defines geometrical and topological functions of skeleton to locate initial critical cutting points, and then employs salient contours with negative minimal principal curvature values to determine natural boundary curves among parts. Sufficient experiments have been carried out on many meshes, and have shown that our framework could provide more perceptual results than pure skeleton-driven or minima-rule-based algorithm.

In a resource-constrained computing environment, it is essential to simplify complex meshes of a huge 3D model for visualization, storing and transmission. Over the past few decades, quadric error metric(QEM) has been the most popular error evaluation method for mesh simplification because of its fast computation time and good quality of approximation. However, quadric based simplification often suffers from its large memory consumption. Since recent 3D scanning systems can acquire both geometry and color data simultaneously, the size of model and memory overhead of quadric increases rapidly due to the additional color attribute. This paper proposes a new error estimation method based on QEM and half-edge collapse for simplifying a triangular mesh model which includes vertex color. Our method calculates geometric error by the original QEM, but reduces the required memory for maintaining color attributes by a new memory-efficient color error evaluation method.

The paper deals with the problem of remeshing and fairing of undersampled areas (called ”holes”) in triangular meshes. In this work, we are particularly interested in meshes constructed with geological data but the method can however be applied to any kind of data. With such input data, the point density is often drastically lesser in some regions than in others: this leads to what we call ”holes”. Once these holes identified, they are filled using a multistep approach. We iteratively: insert vertices in the hole in order to progressively converge towards the density of its neighbourhood, then deform this patch mesh (by minimizing a discrete thin-plate energy) in order to restore the local curvature and guarantee the smoothness of the hole boundary. The main goal of our method is to control both time and space complexity in order to handle huge models while producing quality meshes.

A general formulation based on the variation of digitized energy to denoise image is proposed in this paper. This method is different from classical variational method employed in image processing. For a digitized energy functional, we first compute the variation, then design algorithms leading to digital filters. Numerical experiments and comparative examples are thus carried out to verify the effectiveness of the proposed method, which is efficient, adaptive and easily implemented. Higher quality images can be obtained with characteristic singular features preserved. The method can be easily expanded to multichannel image denoising.

An important surface modeling problem in CAD is to connect two disjoint B-spline patches with the second-order geometric continuity. In this paper we present a study to solve this problem based on the surface extension algorithm [Computer-Aided Design 2002; 34:415–419]. Nice properties of this extension algorithm are exploited in depth and thus make our solution very simple and efficient. Various practical examples are presented to demonstrate the usefulness and efficiency of our presented solution.

The triangular mesh surfaces (TMS) which result form the Marching Cubes (MC) algorithm have some unique and special properties not shared by general TMS. We exploit some of these properties in the development of some new, effective and efficient methods for parameterizing these surfaces. The parameterization consists of a planar triangulation which is isomorphic (maps one-to-one) to the triangular mesh. The parameterization is computed as the solution of a sparse linear system of equations which is based upon the fact that locally the MC surfaces are functions (height-fields). The coefficients of the linear system utilize natural neighbor coordinates (NNC) which depend upon Dirchlet tessellations. While the use of NNC for general TMS can be somewhat computationally expensive and is often done procedurally, for the present case of MC surfaces, we are able to obtain simple and explicit formulas which lead to efficient computational algorithms.

In this paper, parametric polynomial minimal surfaces of degree six with isothermal parameter are discussed. We firstly propose the sufficient and necessary condition of a harmonic polynomial parametric surface of degree six being a minimal surface. Then we obtain two kinds of new minimal surfaces from the condition. The new minimal surfaces have similar properties as Enneper’s minimal surface, such as symmetry, self-intersection and containing straight lines. A new pair of conjugate minimal surfaces is also discovered in this paper. The new minimal surfaces can be represented by tensor product Bézier surface and triangular Bézier surface, and have several shape parameters. We also employ the new minimal surfaces for form-finding problem in membrane structure and present several modeling examples.

This paper describes a stable and robust finite element solver for physically-based texture synthesis over arbitrary manifold surfaces. Our approach solves the reaction-diffusion equation coupled with an anisotropic diffusion equation over surfaces, using a Galerkin based finite element method (FEM). This method avoids distortions and discontinuities often caused by traditional texture mapping techniques, especially for arbitrary manifold surfaces. Several varieties of textures are obtained by selecting different values of control parameters in the governing differential equations, and furthermore enhanced quality textures are generated by fairing out noise in input surface meshes.

In this paper a novel cross correlation technique is proposed for shape matching between two similar objects. The proposed technique can not only evaluate the similarity between any two objects, but also has two distinct advantages compared to previous work: (1) the deformed articulated objects such as human being with different poses, can be matched very well; (2) the local feature extraction and correspondence can be established at the same time. The basic tool we used is the shape profile driven from the curvature map of the object profile. The cross correlation technique is applied to the shape profile of the two objects to evaluate their similarity. Filtering scheme is used to enhance the quality of both shape matching and extracted features. The invariant property, the robustness and the efficiency of the shape profile in shape matching and feature extraction are discussed.

We use a moving parabolic approximation (MPA) to reconstruct a triangular mesh approximating the underlying surface of a point cloud. We suggest an efficient procedure to generate an initial mesh from a point cloud of closed shape. Then we refine this mesh selectively by comparing estimates of curvature from the point cloud with curvatures computed from the current mesh. We present several examples which demonstrate robustness of our method in the presence of noise, and show that the resulting reconstructions preserve geometric detail.

Fitting of data points by parametric curves and surfaces is demanded in many scientific fields. In this paper we review and analyze existing least squares orthogonal distance fitting techniques in a general numerical optimization framework. Two new geometric variant methods (

and

) are proposed. The geometric meanings of existing and modified optimization methods are also revealed.

A degree

n

rational plane curve rotating about an axis in the plane creates a degree 2

n

rational surface. Two formulas are given to generate 2

n

moving planes that follow the surface. These 2

n

moving planes lead to a 2

n

×2

n

implicitization determinant that manifests the geometric revolution algebraically in two aspects. Firstly the moving planes are constructed by successively shifting terms of polynomials from one column to another of a spawning 3×3 determinant. Secondly the right half of the 2

n

×2

n

implicitization determinant is almost an

n

-row rotation of the left half. As an aside, it is observed that rational parametrizations of a surface of revolution due to a symmetric rational generatrix must be improper.

Surface parameterization is a fundamental tool in geometric modeling and processing. Different to most existing methods, this work introduces a

linear

method, called

slit map

, to conformally parameterize multiply connected disk-like surfaces with

canonical

domains, namely

circular slit domain

(annulus with concentric circular slits) or

parallel slit domain

(rectangle with parallel slits) equivalently. The construction is based on holomorphic one-forms, which is intrinsic, automatic and efficient. Slit map owns many merits over existing methods. The regularity of the image boundaries simplifies many geometric process, such as surface matching and etc; the full utilization of the parallel slit texture domain improves the packing efficiency for texture mapping; the positions of the slits are completely determined by the surface geometry and can be treated as the finger prints for surface classification. In the paper, both the underlying theory and the algorithm pipeline are explained. Preliminary experimental results are shown for several application purposes.

We present a new constructive solving algorithm for systems of 3D geometric constraints. The solver is based on the cluster rewriting approach, which can efficiently solve large constraint systems, and incrementally handle changes to a system, but can so far solve only a limited class of problems. The new solving algorithm extends the class of problems that can be solved, while retaining the advantages of the cluster rewriting approach. It rewrites a system of constraints to clusters with various internal degrees of freedom. Whereas previous constructive solvers only determined rigid clusters, we also determine two types of non-rigid clusters. This allows us to solve many additional problems that cannot be decomposed into rigid clusters, without resorting to expensive algebraic solving methods.

This paper presents a method for analogizing high-dimensional space-time curves, and shows how it can be used to transform a motion sequence into new content and styles according to two accompanied reference sequences. By providing examples, our system can estimate the parameters of a statistical model which incorporates spatial and temporal factors for motion curves. A user can then easily modify existing motion sequences, within these parameters, to have new semantics as examples. This method can also be used to enrich motion capture databases or edit recorded animations in batch mode.

This paper considers the problem of computing an interpolating skin of a ordered set of discrete 2D balls. By construction, the skin is constrained to be

C

1

continuous, and for each ball, it touches the ball at a point and is tangent to the ball at the point of contact. Using an energy formulation, we derive differential equations that are designed to minimize the skin’s arc length, curvature, or convex combination of both. Given an initial skin, we update the skin’s parametric representation using the differential equations until convergence occurs. We demonstrate the method’s usefulness in generating interpolating skins of balls of different sizes and in various configurations.

An approach to segmentation of a 3D mesh is proposed. It employs mean-shift curvature to cluster vertices of the mesh. A region-growing scheme is then established for collecting them into connected subgraphs. The mesh faces consisting of vertices in the same subgraph constitute a patch while faces whose vertices are in different subgraphs are split and then lined out to near patches to complete the segmentation. To produce pleasing results, several ingredients are introduced into the segmentation pipeline. Firstly, we enhance the original model before mean-shifting and then transfer the curvature of the enhanced mesh to the original one in order to make the features distinguishable. To rectify the segmentation boundaries, the min-cut algorithm is used to repartition regions around boundaries. We also detect sharp features.

This work is a contribution towards the graphical display of 3D data over a regular grid when it is convex. A piecewise rational bi-cubic function [8] has been utilized for this objective. Simple sufficient data dependent conditions are derived on free parameters in the description of rational bi-cubic function to preserve the shape of data. The presented method applies equally well to data or data with derivatives. The developed scheme is not only local and computationally economical but also visually pleasing.

This paper focuses on an important aspect of cardiac surgical simulation, which is the deformation of mesh models to form smooth joins between them. A novel algorithm based on the Laplacian deformation method is developed. It extends the Laplacian method to handle deformation of 2-manifold mesh models with 1-D boundaries, and joining of 1-D boundaries to form smooth joins. Test results show that the algorithm can produce a variety of smooth joins common in cardiac surgeries, and it is efficient for practical applications.

The paper presents an adaptive slicing algorithm of functionally graded material (FGM) components for RP. The algorithm is studied by considering both geometry information and material distribution information. The layer thickness is computed according to the area variation rate between two adjacent slices, the gradient of material distribution and material resolution. After slicing, the material information in each slice is stored discretely by material composition function and material resolution. Finally, an example is presented.

Personalize mannequin design are the basic element in virtual garment simulation and visualization. This paper addresses the problem of mannequin construction from scanned point cloud and regenerates their shapes by inputting variant dimension. Layer-based simplification and parameterization algorithm is presented to preserve the mannequin’s features as far as possible. The regenerated models are watertight and used for clothes design.

It is well known that the geometric structure of a protein is an important factor to determine its functions. In particular, the atoms located at the boundary of a protein are more important since various physicochemical reactions happen in the boundary of the protein. The

β

-shape is a powerful tool for the analysis of atoms located at the boundary since it provides the complete information of the proximity among these atoms. However,

β

-shapes are difficult to handle and require heavy weight data structures since they form non-manifold structure. In this paper, we propose topology operators for converting a

β

-shape into a manifold. Once it is converted, compact data structures for representing a manifold are available. In addition, general topology operators used for manifold structures can also be available for various applications.

This paper proposes a mesh simplification method by finding the optimal positions of vertices. It generates a Bézier patch around the collapsed edge and finds the optimal position based on visual importance and curvature. It successfully maintains the geometry and topology of the model even when the size of the model is reduced to less than 5 % of the original model. Our method uses QEM for the error measure. It can be applied to usual mesh simplification but also to mesh parameterization and remeshing.

We propose a narrow-band based RBFs implicit surface reconstruction method which can substantially reduce the computational complexity compared with other RBFs implicit surface reconstruction techniques. Our scheme only deals with a narrow-band subdomains, rather than the traditional whole computational domain. A criteria for polygonization is presented for correctly extracting iso-surfaces from RBFs implicits. Experiments show that our method can offer a very effective RBFs based surface reconstruction algorithm.

A new method for constructing interpolating Loop subdivision surfaces is presented. The new method is an extension of the progressive interpolation technique for B-splines. Given a triangular mesh

M

, the idea is to iteratively upgrade the vertices of

M

to generate a new control mesh

$\bar M$

such that limit surface of

$\bar M$

interpolates

M

. It can be shown that the iterative process is convergent for Loop subdivision surfaces. Hence, the method is well-defined. The new method has the advantages of both a local method and a global method, i.e., it can handle meshes of any size and any topology while generating smooth interpolating subdivision surfaces that faithfully resemble the shape of the given meshes.

A PDE (partial differential equation) based method, 3D active contour model, is presented to model the surface of protein structure. Instead of generating a single molecular surface, we create a series of surfaces associated with the atomic energy inside the protein, which describe different resolutions of molecular surface. Our results indicate that the surfaces we generated are suitable for shape analysis and visualization. So, when the solvent-accessible surface is not enough to represent the features of protein structure, the evolution surface sequence may be an alternative choice. Besides, if the initial surface is smooth enough, the generated surfaces will preserve this property partly, because the evolution of the surfaces is controlled by the PDE.

Quasi-interpolation by the radial basis functions is discussed in this paper. We construct the approximate interpolant with Gaussion function. The suitable value of the shape parameter is suggested. The given approximate interpolants can approximately interpolate, with arbitrary precision, any set of distinct data in one or several dimensions. They can approximate the corresponding exact interpolants with the same radial basis functions. The given method is simple without solving a linear system. Numerical examples show that the given method is effective.

Although deforming surfaces are frequently used in numerous domains, only few works have been proposed until now for simplifying such data. In this paper, we propose a new method for generating progressive deforming meshes based on shape feature analysis and deformation area preservation. By computing the curvature and torsion of each vertex in the original model, we add the shape feature factor to its quadric error metric when calculating each QEM edge collapse cost. In order to preserve the areas with large deformation, we add deformation degree weight to the aggregated quadric errors when computing the unified edge contraction sequence. Finally, the edge contraction order is slightly adjusted to further reduce the geometric distortion for each frame. Our approach is fast, easy to implement, and as a result good quality dynamic approximations with well-preserved fine details can be generated at any given frame.

Each protein tertiary structure is represented by three sorts of geometric models, which are polyline curves, triangulated surfaces and volumetric solids. Moment invariants are employed to describe the shapes of the three kinds of protein models and form a multidimensional feature vector for each model. We further analyze the influence of the three representations in applications to protein shape retrieval and classification.

The V-system on

L

2

[0,1]constructed in 2005 is a complete orthogonal system. It has multiresolution property. This paper further studies the V-system of two variables. The orthogonal V-system of degree

k

defined over triangulated domains is presented. With the orthogonal V-system over triangulated domains, all the application of the V-system on

L

2

[0,1] can be generalized onto the surface. Especially, the triangulated surface represented by piecewise polynomial of two variables of degree

k

with multi-levels discontinuities can be precisely reconstructed by finite terms of the V-series.

This paper proposes a novel computation method for tool path planning in 5-axis flank milling of ruled surfaces. This method converts the path planning (a geometry problem) into a curve matching task (a mathematical programming problem). Discrete dynamic programming techniques are applied to obtain the optimal matching with machining error as the objective function. Each matching line corresponds to a cutter location and the tool orientation at the location. An approximating method based on z-buffer is developed for a quick estimation of the error. A set of parameters is allowed to vary in the optimization, thus generating the optimal tool paths in different conditions. They reveal useful insights into design of the tool motion pattern with respect to the surface geometry. The simulation result of machining different surfaces validates the proposed method. This work provides an effective systematic approach to precise error control in 5-axis flank milling.

The problem of trimming Bézier surfaces on Bézier surfaces contains many cases, such as the subdivision, conversion and conjoining. Different methods have been given for some special cases. In this paper, by means of blossoming and parameter transformation, a united approach is given for this problem. The approach can be extended to trim Bézier patches on any polynomial or rational surfaces naturally.

This paper presents a novel unified solution – a general volumetric framework for 3D modeling, interaction, and visualization applications with complex volumetric scenes composed of virtually any types of conventional and unconventional object representations, potentially for a wide range of volumetric applications. As an example, this framework is applied to a general problem of volume modeling, and the experimental results demonstrate the effectiveness, flexibility and generality of our volume fusion algorithms for 3D interactive volumetric applications.

A novel method based on subdivision mesh for geometric calibration problem is proposed, which can correct the projector imagery on smooth curved screen with respect to the screen geometry and the observer’s position. Using sparse initial mesh extracted from the image of camera and a special designed subdivision algorithm, we can subdivide the initial mesh into arbitrary precision. Hence, the one-one correspondence of pixels between the frame buffer of projector and image of camera can be established. Using this correspondence, we can transform the projector imagery into any shape through the image of camera. So the geometric calibration can be done easily by mapping all pixels of frame-buffer image to an undistorted mesh on curved screen specified by a laser theodolite. Theoretical analysis and experimental results show that the method can automatically, accurately and rapidly display an undistorted image on curved screen, without any projector and camera registrations or prior knowledge of screen.

In [2] we presented an algorithm for decomposing a boundary representation model hierarchically into regularity features by recovering broken symmetries. The algorithm adds new recoverable edges and faces, which can be constructed from existing geometry. This generates positive and negative volumes giving simple, more symmetric sub-parts of the model. The resulting regularity feature tree may be utilised for regularity detection to describe a model’s design intent in terms of regularities such as symmetries and congruencies.