Takashi Maekawa

• Yokohama National University

We present a novel approach to generate developable strips along a space curve. The key idea of the new method is to use the rotation angle between the Frenet frame of the input space curve, and its Darboux frame of the curve on the resulting developable strip as a free design parameter, thereby revolving the strip around the tangential axis of the...

A doubly curved surface can be approximated with a piecewise developable surface by constructing the envelope of the family of the tangent planes along given curves on the input surface. In this paper, we utilize this fact to approximate a doubly curved input surface by a piecewise ruled B-spline surface with high developability. This brings advant...

We propose a new approach for the accurate numerical computation of high-order derivatives along geodesic curves on surfaces. The method is based on the observation that for geodesics the Darboux frame and the Frenet–Serret frame are locally equal up to a constant rotation around the tangent. It computes derivatives of arbitrary order from the resu...

We introduce a fabrication method of doubly-curved shell structures by CFRP with control over fiber directions. Doubly-curved surfaces are tessellated into structured quadrilateral patches according to the purpose of use. One can design the tessellation in the parameter space, and map it to the 3D parametric surface, or use the orthogonal net of cu...

We propose an approach to generate geodesic grids for approximating freeform surfaces. We employ the accurate numerical computation of high-order derivatives along geodesic curves on surfaces to determine the width of the geodesic strips. It computes derivatives of arbitrary order from the result of the numerical method employed for computing the g...

We show that the surface of an umbrella has negative Gaussian curvature, and due to its symmetry, the surface of the umbrella between bones can be modeled as a piecewise bilinear surface. We investigate various differential geometric properties of the umbrella surface and demonstrate the effectiveness of our proposed geometric modeling of umbrella...

Mostly, umbrellas are designed empirically. We consider using numerical analysis for an umbrella design. We use T-spline to represent the model to have smooth basis functions as a mesh for structural mechanics. The challenge is to represent the ribs as a one-dimensional model on a cloth represented by a smooth basis function. We use the T-spline fo...

We present a two-lane path planning of autonomous vehicles in three-dimensional (3D) environments. Our method builds on collision-free two-dimensional (2D) path-planning techniques from our previous work, and extends these techniques to three dimensions by mapping a 2D path in the parameter space onto a terrain graph surface, as a curve on the surf...

A ship hull consists of smooth streamlined surfaces satisfying hydrodynamic characteristics such as frictional, pressure, and wave resistances. It contains doubly curved surfaces particularly at the bow and stern. In shipbuilding, surface plates are formed by the mixture of cold and hot bending, which is typically called line heating. An efficient...

Recently, demand for the use of B-spline surfaces to fit point clouds by geometric iterative method (GIM) is increasing due to its simplicity, scalability, and generality. Unlike conventional method, GIM fits point clouds by repositioning control points iteratively on the basis of geometrical rules. In this paper, we propose novel methods for a sur...

Recent advance of Additive Manufacturing technologies allows us to manufacture various parts used in real-world products. Consequently, product tracking of such 3D printed parts is an important issue. Quick Response (QR) code which is a two-dimensional matrix barcode invented by Denso, a Japanese automotive industry, in 1994, can be used for this p...

Reverse engineering of 3D industrial objects such as automobiles and electric appliances is typically performed by fitting B-spline surfaces to scanned point cloud data with a fairing term to ensure smoothness, which often smooths out sharp features. This paper proposes a radically different approach to constructing fair B-spline surfaces, which co...

In recent years, 3D printers have been utilized in many fields such as aviation industry, microelectromechanical systems and medicine. The size of 3D printed object is ranging from several micrometers to several meters. In particular, microstereolithography makes it possible to produce 3D micro objects whose size is less than a few centimeters. In...

3D printers can easily create complex shapes and now are becoming popular due to their price dropping. Since various shapes can be generated by assembling block parts like LEGO, a block type 3D printer can also be realized. We study a robot system “Block Printing System” that automatically performs assembly of blocks from 3D CAD models. In our prev...

This study aims to assemble microblocks with a robot by manual and automatic operation. We use a SCARA-like robot and a three-axis stage, which are driven by ultrasonic motors. The three-axis stage changes position and orientation of the microblocks so that they can be picked up by the SCARA-like robot. A camera is installed above the stage to obse...

Geometric iterative methods (GIM), including the progressive–iterative approximation (PIA) and the geometric interpolation/approximation method, are a class of iterative methods for fitting curves and surfaces with clear geometric meanings. In this paper, we provide an overview of the interpolatory and approximate geometric iteration methods, prese...

We introduce a framework for modeling of heterogeneous objects in terms of trivariate B-spline functions and a method for slicing them directly for additive manufacturing. We first fit volumetric attribute data associated with the geometry in terms of trivariate B-spline functions under the assumption that the geometric volume is already defined by...

In this paper we introduce an algorithm for generating collision-free two-lane haul road for unmanned vehicles used in the mining areas in terms of quartic B-spline curves. Given the boundary geometry of the haul road area, positions and 2D vehicle orientation at the start and goal points, the algorithm automatically generates collision-free haul r...

In recent years, 3D printers have been utilized in many fields. The size of 3D printed object is ranging from several micrometers to several meters. In particular, microstereolithography makes it possible to produce 3D micro objects whose size is less than a few centimeters. In this process, photopolymer is selectively polymerized by scanning a las...

Current CAD modeling techniques enable the design of objects with aesthetically pleasing smooth freeform surfaces. However, the fabrication of these freeform shapes remains challenging. Our novel method uses orthogonal principal strips to fabricate objects whose boundary consists of freeform surfaces. This approach not only lends an artistic touch...

We study the effects of curvature on the energy absorption characteristics of cylindrical corrugated tubes under compression by isogeometric analysis and experiments. The corrugated volume is constructed by revolving a wavelike B-spline profile surface about the vertical axis. The curvature at peaks of the profile curve is gradually increased from...

When drilling circular holes into metal circular pipes, burr is generated at the hole entrance as well as at the hole exit. The burrs generated at the edge curves associated with the outer and inner pipe surfaces must be removed by constant chamfering. Geometrically, the two edge curves can be defined as cylinder-to-pipe intersection curves. In thi...

In the manufacturing process of machine products, it is important to automate the manual visual inspection in detecting microscopic surface defects in order to improve the efficiency and eliminate human errors as well. The basic hardware system consists of a high-resolution camera equipped with a telecentric lens, and a device to change light direc...

We present a method to face hydraulic excavator attachments parallel to target surfaces by determining the boom, arm, bucket, and tilt angles of the manipulator. In order to determine these four angles, Powell’s method is employed to find a local minimum of the objective function where the evaluation of the derivatives is not required. Examples are...

Direct laser writing (DLW) based on two-photon microstereolithography has been used for many kinds of application fields such as MEMS and lab-on-a-chip. In this study, we have developed an ultrafast multiscale two-photon microstereolithography system by using spatial light modulation technique. An optical vortex generated by a spatial light modulat...

In this study, we propose a novel method named "3D block printing" to make a 3D object with different materials. This system allows us to produce arbitrary shaped 3D models by assembling multiple blocks of different materials. For medical application, bioceramic blocks for constructing 3D scaffolds were made with β-tricalciumphosphate (β-TCP) slurr...

We introduce a method to generate developed paper craft patterns of free-form surfaces based on lines of curvature. Objects that consist of free-form surfaces can be developed onto a plane by expanding the surfaces along the lines of curvature. If we remove all the expanded areas, paper crafts can be constructed by connecting the corresponding edge...

Finite element analysis and NURBS-based computer-aided design had been developed independently until isogeometric analysis (IgA) was introduced by Hughes et al. in 2005. Unlike FEA, in isogeometric analysis the CAD geometry represented by NURBS curves/surfaces/volumes are converted to NURBS elements exactly without any approximations. On the other...

This paper studies an automatic 3D shape reconstruction of objects using images taken by a camera from different views based on photometric stereo technique. The automated data acquisition system consists of a camera, lighting system that consists of 60 LED bulbs, and a turntable rotated by a computer-controlled stepping motor. We use photometric s...

In the manufacturing process of machine products, it is very important to automate the manual visual inspection in detecting microscopic surface defects in order to improve the efficiency and eliminate human errors as well. We present an automatic system to detect such surface defects based on iterative photometric stereo techniques. Our system con...

We introduce a system to reconstruct large scale LEGO models from multiple two dimensional images of objects taken from different views. We employ a unit voxel with an edge length ratio of 5:5:6 for the shape from silhouette method that reconstructs an octree voxel-based three dimensional model with color information from images. We then convert th...

We introduce a system to reconstruct LEGO models from multiple two-dimensional images of complex objects taken from different views based on shape from silhouettes (SFS) technique. We employ a unit voxel with edge length ratios 5:5:6 for the SFS method, and convert the resulting voxel models with color information to LEGO models. In order to minimi...

This paper studies the recovery of the height field function of an object from its surface normal vectors in terms of uniform bi-quadratic B-spline surfaces. The study is motivated by two modern technological needs, namely the recovery of the height field function of an object from its surface normal vectors obtained by photometric stereo technique...

This paper presents a novel method for modifying the shapes of existing uniform bi-cubic B-spline surfaces by interactively editing the curvatures along isoparametric curves. The method allows us to edit the curvatures of the two intersecting isoparametric curves at each knot with specified positions, unit tangents, and unit normals. The user adjus...

In the past, tremendous amount of research has been conducted to fit point clouds using B-spline curves and surfaces. However, there are few studies on fitting scientific volume data using the trivariate B-spline functions. In this paper, an iterative geometric approximation method, which iteratively repositions B-spline control points on the basis...

It is an important task to generate an efficient operation planning for a modern large scale open-pit mine. Given the depth and the boundary curves of the mine distributions from the boring survey, we introduce a novel method to design the geometry of an open-pit mine. As the first step, the region of interest including the boundary curves of the m...

An approximation processing method for approximating a point group using a curve or a surface defined by control points includes a step of forming an approximated curve (surface) using control points that retain features of a shape; a first calculation step of calculating a closest point closest to each of the data points on the approximated curve...

Finite element analysis (FEA) and NURBS-based computer-aided design (CAD) had been developed independently until isogeometric analysis, was introduced by Hughes et al in 2005. Unlike FEA, in isogeometric analysis the CAD geometry represented by NURBS surfaces/solids are converted to NURBS elements exactly without any approximations In this paper we...

We herein propose a novel method for removing irregularities of B-spline surfaces via smoothing circular highlight lines. A circular highlight line is defined as a set of points on a surface such that the distance between a circular light source and an extended surface normal to be zero. Circular highlight lines allow us to capture the surface fair...

We present a novel method to assess the fairness of class A surfaces. The fairness of class A surfaces are often assessed by reflection curves on the surfaces of a family of parallel straight fluorescent lights, namely reflection lines and highlight lines. Unlike reflection lines and highlight lines, where a family of parallel straight lines are us...

A method for reconstructing a complete three-dimensional (3D) voxel model from multiple images captured during two circular motions of an object is proposed. The two circular motions include the object with a regular orientation and the corresponding reversed orientation. The shape-from-silhouette method, which constructs a voxel-based 3D model fro...

A travel route generating method for an unmanned vehicle, which includes a condition input step of inputting a vehicle constraint condition including a vehicle width and a minimum turning radius of the unmanned vehicle, and a geometrical constraint condition including a travel route generation range in which the unmanned vehicle is to travel, an ob...

When an interpolation point is interpolated using a curve/curved surface, represented by control points, such as a B-Spline curve/curved surface or a subdivision surface, interpolation processing is performed in such a way that a moving vector for moving each control point is calculated using position information on interpolation points constitutin...

In this paper, we present a procedure for automatically reconstructing an arbitrary topological surface from an unorganized point data set; this surface will have three representations, namely quadrilateral meshes, Catmull–Clark subdivision surfaces, and B-spline surfaces. Our novel reconstruction method adapts a level set method to capture the top...

We present a method for reconstructing complete 3D models from a number of images of an object taken from multiple viewpoints under the circular motions. We employ the shape from silhouette (SFS) method which constructs a voxel-based 3D model from the silhouettes of 2D images. In general, a single camera with an object's circular motion cannot capt...

Recently, the advancement of the laser range scanner has made it possible to provide dense and accurate range data of 3D objects. From these point clouds, one can reconstruct a CAD model of the existing old mechanical parts, which is called reverse engineering. However, if we fit the point cloud without any constraints on geometric characteristics,...

Recently, the use of BB-spline curves/surfaces to fit point clouds by iteratively repositioning the BB-spline’s control points on the basis of geometrical rules has gained in popularity because of its simplicity, scalability, and generality. We distinguish between two types of fitting, interpolation and approximation. Interpolation generates a BB-s...

In many engineering applications, a smooth surface is often approximated by a mesh of polygons. In a number of downstream applications, it is frequently necessary to estimate the differential invariant properties of the underlying smooth surfaces of the mesh. Such applications include first-order surface interrogation methods that entail the use of...

In this paper we discuss the self-intersections of offset curves and surfaces and show how to determine the maximum offset distance such that the offset does not locally nor globally self-intersect including boundary effects. Examples illustrate the applicability of the analysis.

In this paper, we propose a novel method for approximating a set of scattered data points by a quadrilateral mesh using level set methods followed by geometric algorithms. The level set method is used to capture the topology of the point clouds, and the geometric algorithm using Catmull-Clark subdivision surface is employed to generate high quality...

In this research, we introduce a novel method to analyze the axial compression analysis of thin circular shell structures. employing a new analysis framework, called isogeometric analysis, which is based on NURBS surfaces We generate the thin circular shell structures by first designing the profile curve with prescribed positions, normals and curva...

In the current highway design, a center curve of a highway is designed in three steps. In the first step the projection of the center curve onto the horizontal plane is designed with planar curves, i.e. x=x(t), y=y(t) consisting of straight lines, circular arcs, and clothoid curves. Secondly, the elevation z=z(t) is designed separately. Finally, th...

This paper presents a geometric algorithm for the generation of uniform cubic B-spline curves interpolating a sequence of data points under tangent and curvature vectors constraints. To satisfy these constraints, knot insertion is used to generate additional control points which are progressively repositioned using corresponding geometric rules. Co...

We introduce a system to reconstruct a three-dimensiojnal (3D) polygonal model of 3D micro objects with outer dimensions ranging from several hundred microns to several millimeters from multiple two-dimensional (2D) images of an object taken from different views. The data acquisition system consists of a digital microscope that captures still image...

We introduce a new method to approximate a point cloud by B-spline surfaces by the accelerated geometric algorithm. We distinguish two types of fitting, interpolation and approximation. In this paper we present a method to approximate B-spline surfaces that pass near point clouds. Our method takes an approach that the initial shape is updated step...

3D Objects in computer graphics are often represented by triangular meshes. Even surfaces are represented by piecewise planar meshes, we are often required to evaluate differential geometry properties such as normal vectors, principal curvatures, principal directions, locations of umbilics of the underlying smooth surface. However, there is no stan...

We first introduce three forms to represent geometric objects mathematically. They are the parametric, implicit and explicit forms. Implicit and explicit forms are often referred to as nonparametric forms. Then we briefly review the representation of curves and surfaces in Bézier and B-spline form and treat the special properties associated with ea...

An umbilic is a point on a surface where all normal curvatures are equal in all directions, and hence principal directions are indeterminate. Thus the orthogonal net of lines of curvature, which is described in Sect. 3.4, becomes singular at an umbilic. An obvious example of a surface consisting entirely of umbilical points is the sphere. Actually,...

The computation of maximal and minimal distance between points sets in Euclidean space is a basic problem in computational geometry and geometric modeling. It is useful in surface intersections [210], numerical control machining, tolerance region and access space representation in solid modeling, robotics, inspection of manufactured objects [353, 2...

This paper presents a constructive method for generating a uniform cubic B-spline curve interpolating a set of data points simultaneously controlled by normal and curvature constraints. By comparison, currently published methods have addressed one or two of those constraints (point, normal or cross-curvature interpolation), but not all three constr...

In this paper we introduce a roadmap algorithm for generating collision-free paths in terms of cubic B-spline curves for unmanned vehicles used in mining operations. The algorithm automatically generates collision-free paths that are curvature continuous with an upper bounded curvature and a small slope discontinuity of curvature at knots, when we...

The differential geometry of curves and surfaces is fundamental in Computer Aided Geometric Design (CAGD). The curves and surfaces treated in differential geomentry are defined by functions which can be differentiated a certain number of times. Books by Hilbert and Cohn-Vossen [165], Koenderink [205] provide intuitive introductions to the extensive...

Offset curves/surfaces, also called parallel curves/surfaces, are defined as the locus of the points which are at constant distant d along the normal from the generator curves/surfaces. A literature survey on offset curves and surfaces was carried out by Pham [313] and more recently by Maekawa [250]. Offsets are widely used in various applications,...

We have seen in Chap. 1 that curves and surfaces in CAD/CAM systems are usually represented by piecewise polynomial equation of various type. As we will see in the remaining chapters of this book, the governing equations for general interrogation problems on such curve and surface representations (intersections, distance functions, curvature extrem...

Shape interrogation is the process of extraction of information from a geometric model. It is a fundamental component of Computer Aided Design and Manufacturing (CAD/CAM) systems. The authors focus on shape interrogation of geometric models bounded by free-form surfaces. Free-form surfaces, also called sculptured surfaces, are widely used in the bo...

Computation of shortest paths on free-form surfaces is an important problem in geometric design of ship hulls, robot motion planning, computation of medial axis transforms of trimmed surface patches, terrain navigation, NC machining, and cable installation on the sea floor. The history of geodesic lines begins with a study by Johann Bernoulli, who...

Let us consider a curve u = u(t), v = v(t) in the parametric domain of a parametric surface r = r(u, v) as shown in Fig. 3.1. Then r = r(t) = r (u(t), v(t)) is a parametric curve lying on the surface r = r(u,v).The tangent vector to the curve on the surface is evaluated by differentiating r(t) with respect to the parameter t using the chain rule an...

Intersections are fundamental in computational geometry, geometric modeling and design, analysis and manufacturing applications [276,19, 175, 295, 300]. Examples of intersection problems include: Contouring of surfaces through intersection with a series of parallel planes or coaxial circular cylinders or cones for visualization (see Fig. 5.1). Nume...

In this paper we compare two methods for interpolating a triangular mesh of arbitrary topological type by a Loop subdivision surface. The first method is a conventional interpolation method that solves a linear system, while the second method is based on a geometric algorithm which iteratively updates the control mesh in a global manner based on a...

B-spline surface interpolation technique by geometric algorithm iteratively updates the control mesh in a global manner based on a simple point-surface distance computation followed by translations of the control vertices along the error vectors. In this paper we introduce two techniques to accelerate the computation of geometric interpolation algo...

We introduce a novel method to interpolate a set of data points as well as unit tangent vectors or unit normal vectors at the data points by means of a B-spline curve interpolation technique using geometric algorithms. The advantages of our algorithm are that it has a compact representation, it does not require the magnitudes of the tangent vectors...

We present a method of reconstructing a 3D model from several 2D images of an object taken from different views. The data acquisition system consists of a camera on a tripod, and a computer-controlled turntable. We employ the Shape-From-Silhouette (SFS) method, which constructs a voxel-based 3D model from silhouette images. The concave shapes are f...

Recently, mining industries are actively introducing autonomous dump truck technology in the area to reduce accidents caused by fatigue of the driver, and to cut labor costs. The paths for autonomous vehicles are usually made up of clothoid arcs or polynomial curves connected by tangential line segments. The curvature profile of such path is contin...

This paper describes a method to approximate point sets by Loop subdivision surfaces based on geometric algorithms. We assume that the data points are given in triangular mesh of arbitrary topological type. The initial control mesh of the Loop subdivision surface is obtained by simplifying the input triangular mesh using QEM algorithm. Our algorith...

We present a method to reconstruct a voxel-based 3D model from several 2D images of an object of different views. The data acquisition system consists of a camera on a tripod, and a computer controlled turntable. We employ a so called Shape-From-Silhouette (SFS) method which constructs a voxel-based 3D model from silhouette images. The obtained vox...

We present a method to evaluate the shape of 3D micro products from several images. The data acquisition system consists of a digital microscope, and a computer controlled turntable. We employ a so called Shape-From-Silhouette (SFS) method which constructs a voxel-based 3D model from silhouette images. The obtained voxel model is further carved by...

We present a method for solving arbitrary systems of N nonlinear polynomials in n variables over an n-dimensional simplicial domain based on polynomial representation in the barycentric Bernstein basis and subdivision. The roots are approximated to arbitrary precision by iteratively constructing a series of smaller bounding simplices. We use geomet...

We present a novel geometric algorithm to construct piecewise smooth surface models based on subdivision that interpolates a triangular mesh of arbitrary topological type. All sharp features such as creases and corners of the original mesh model are first identified, and then we assume that the given triangular mesh is a control net of the subdivis...

We present a method to reconstruct a voxel-based 3D model from several 2D images of an object of different views. The data acquisition system consists of a camera on a tripod, and a computer controlled turntable. We employ a so called Shape-From-Silhouette (SFS) method which constructs a voxel-based 3D model from silhouette images. The obtained vox...

Interpolation is one of the most typical problems that arise in the application of B-spline curves and surfaces. In this paper we are interested in not only interpolating a set of points, but also normal vectors given at these points by the B-spline curves. Applications of point-normal fitting include design of functional surfaces used in hydrodyna...

In numerous instances, accurate algorithms for approximating the original geometry is required. One typical example is a circle involute curve which represents the underlying geometry behind a gear tooth. The circle involute curves are by definition transcendental and cannot be expressed by algebraic equations, and hence it cannot be directly incor...

We present a novel geometric algorithm to construct a smooth surface that interpolates a triangular or a quadrilateral mesh of arbitrary topological type formed by n vertices. Although our method can be applied to B-spline surfaces and subdivision surfaces of all kinds, we illustrate our algorithm focusing on Loop subdivision surfaces as most of th...

The study of smooth surface reconstruction from a set of 3D range data obtained by a laser scanner has become increasingly important in the area of reverse engineering, animation, and scientific visualization. The most popular representation for fitting such data in geometric modeling is the tensor product B-Spline surfaces. However, it is extremel...

The purpose of this study is to introduce a new method to fit a set of point by B-spline curves/surfaces by geometric algorithms. Standard B-spline curve/surface fitting algorithms first associate suitable parameter values for each input point, known as parameterization of data, and form a linear system with control points as unknowns. In this pape...

We propose a novel method to remove irregularities of B-spline surfaces via smoothing circular highlight lines. A circular highlight line is defined as a set of points on a surface where the distance between a circular light and an extended surface normal at the circular highlight line is zero. It is very sensitive to the change of the directions o...

Link: http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&p=1&u=%2Fnetahtml%2FPTO%2Fsearch-bool.html&r=1&f=G&l=50&co1=AND&d=PTXT&s1=6,956,568.PN.&OS=PN/6,956,568&RS= Umbilics of two surfaces are compared and it is determined from this comparison whether the suspect surface is a copy of the original surface based on the comparison. C...

A free-form object matching problem is addressed in this paper. Two methods are proposed to solve a partial matching problem with scaling effects and no prior information on correspondence or the rigid body transformation involved. The first method uses umbilical points, which behave as fingerprints of a surface and their qualitative properties can...

Recently copyright issues for digital contents are becoming a serious problem. Especially when the copyrighted digital contents are exposed to the internet, they are an easy target for malicious parties to produce pirate digital contents for unauthorized sales. Digital watermarking, defined as a process to embed data called watermark into a digital...

This paper presents a parametric and feature-based methodology for the design of solids with local composition control (LCC). A suite of composition design features are conceptualized and implemented. The designer can use them singly or in combination, to specify the composition of complex components. Each material composition design feature relate...

This paper presents a robust method for tracing intersection curve segments between continuous rational parametric surfaces, typically rational polynomial parametric surface patches. The tracing procedure is based on a validated ordinary differential equation (ODE) system solver which can be applied, without substantial overhead, for transversal as...

The Boolean algebra of regular closed sets is prominent in topology, particularly as a dual for the Stone-Cech compactification. This algebra is also central for the theory of geometric computation, as a representation for combinatorial operations on geometric sets. However, the issue of computational approximation introduces unresolved subtleties...

This paper presents an algorithm of modifying free-formed NURBS curve/surface for offsetting without local self-intersecting. The method consists of (1) sampling a number of points from a progenitor curve/surface based on second derivatives; (2) checking the curvature or maximum curvature of the progenitor curve/surface at the sampled points; (3) i...

The Boolean algebra of regular closed sets is prominent in topol- ogy, particularly as a dual for the Stone- Cech compactication. This algebra is also central for the theory of geometric computation, as a representation for combinatorial operations on geometric sets. However, the issue of com- putational approximation introduces unresolved subtleti...

This paper presents an overview of surface intersection problems and focuses on the rational polynomial parametric/rational polynomial parametric surface intersection case including transversal and tangential intersections. Emphasis is placed on marching methods with a discussion of the problems with conventional tracing algorithms. An approach usi...

This paper presents matching and similarity evaluation meth- ods between two NURBS surfaces, and their application to copyright protection of digital data representing solids or NURBS surfaces. Two methods are employed to match ob- jects: the moment and the curvature methods. The moment method uses integral properties, i.e. the volume, the princi-...