;------------------------------------------------------------------------------- Computer-aided Design, Vol. 29 (5) (1997) pp. 379-391 © 1997 Elsevier Science B.V. All rights reserved. PII: S0010-4485(96)00077-2 Robust computation of the rotation minimizing frame for sweep surface modeling Wang Wenping and Barry Joea a Department of Computing Science, University of Alberta, Edmonton, Alberta, T6G 2H1, Canada Received 17 February 1996; received 09 August 1996 Abstract The rotation minimizing frame is superior to the Frenet frame for modeling sweep surfaces [F. Klok, Computer Aided Geometric Design 3, 217-229 (1986)]. However, the existing techniques for computing the rotation minimizing frame either have low approximation degree or are unrobust numerically. We present a method to compute an approximate rotation minimizing frame in a robust and efficient manner. The following problem is studied. Given an axial curve A(u) in space and a 2D cross-section curve C(v), generate a sweep surface S(u, v) = A(u) + F(u)C(v), where F(u) is a rotation minimizing frame defined on A(u). Our method works by approximating A(u) with a G1 circular-arc spline curve and then sweeping C(v) with a rotation minimizing frame along the approximating circular-arc spline curve; the sweep surface thus generated is an approximation of S(u, v). The advantages of this method are: (1) the approximate rotation minimizing frame is computed robustly, with its error being much smaller than would be obtained by Klok's linear method with the same number of segmentations; (2) the sweep surface generated is a NURBS surface if the cross-section curve is a NURBS curve; (3) the method is easily adapted to generating a smooth and closed sweep surface when A(u) is a closed smooth curve. Keywords: surface modeling; biarc; NURBS; sweep surface; moving frame ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 29 (5) (1997) pp. 361-368 © 1997 Elsevier Science B.V. All rights reserved. PII: S0010-4485(96)00074-7 Symbolic operators for NURBS Les Piegl * and Wayne Tillera a GeomWare, Inc., 3036 Ridgetop Road, Tyler, TX 75703, USA Received 15 March 1996; received 30 July 1996 Abstract Symbolic operators for NURBS curves and surfaces are presented in this paper. The operators are used to compute NURBS entities by performing algebraic operations using NURBS curves and surfaces as variables. Dot and cross products, sum/difference and derivative operators are presented. An application to construct ruled surfaces to rational rail curves is also included. Keywords: NURBS; symbolic operators; geometric algorithms *Corresponding author © Copyright 2001, Elsevier Science, All rights reserved. ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 28 (9) (1996) pp. 699-706 © 1996 Elsevier Science B.V. All rights reserved. PII: S0010-4485(95)00084-4 Algorithm for approximate NURBS skinning Les Pieglb and Wayne Tillera b Department of Computer Science and Engineering, University of South Florida, 4202 Fowler Avenue, ENG 118, Tampa, FL 33620, USAa GeomWare, Inc., 3036 Ridgetop Road, Tyler, TX 75703, USA Received 14 August 1995; received 10 November 1995 Abstract An algorithm for approximate skinning through cross-sectionalNURBS curves is presented. The method eliminates the problem of dealing with huge amounts of control points obtained during the curve compatability process. It also allows the designer to specify large numbers of cross-sections and approximately fit a smooth surface to these curves to any given tolerance. Depending on the tolerances used, up to 99% of the control points can be eliminated. Keywords: NURBS; surface skinning; curves and surfaces; algorithms ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 28 (6-7) (1996) pp. 461-472 © 1996 Elsevier Science B.V. All rights reserved. PII: S0010-4485(95)00043-7 A tessellation algorithm for the representation of trimmed NURBS surfaces with arbitrary trimming curves Bernd Hamann Tsai Po-Yua Department of Computer Science, University of California, Davis, CA 95616-8562, USAa NSF Engineering Research Center for Computational Field Simulation, Mississippi State University, PO Box 6176, Mississippi State, MS 39762, USA Received 28 February 1995; received 30 April 1995 Abstract In most CAD systems, a trimmed parametric surface is defined by two things: the control information of the surface itself and the control information of its trimming curves, which are usually defined as parametric curves in the parameter space of the surface. Often, trimmed parametric surfaces cause problems in the context of data exchange between CAD systems, surface evaluation/rendering, and grid/mesh generation. This paper describes a new approach for representing a trimmed surface. The approach is based on the idea of decomposing the valid part of the parameter space of a trimmed surface by a set of planar, ruled surfaces, which are defined in parameter space and whose union defines the entire valid part. A generalized Voronoď diagram is constructed for the set of trimming curves in parameter space. This diagram defines tiles around each trimming curve, which are further tessellated by utilizing a scan line technique for identifying and filling the interior of polygons. Here, the scan line technique is used to extract non-horizontal segments of the given trimming curves. Pairs of these segments are then linearly interpolated in horizontal parameter direction, thus defining a set of planar, ruled surfaces. The set of all these planar, ruled surfaces defines the entire valid part of a trimmed surface exactly. Keywords: bisector; NURBS; scan-line algorithm; trimmed surface; Voronoď diagram ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 27 (1) (1995) pp. 16-26 © 1995 Elsevier Science B.V. All rights reserved. PII: S0010-4485(95)90749-6 Tessellating trimmed nurbs surfaces Leslie A Piegl and Arnaud M Richard Received 09 September 1993; received 01 December 1994 Abstract An algorithm for obtaining a piecewise planar approximation of a trimmed nurbs surface is presented. Given a model space tolerance [ϵ], the algorithm triangulates the parameter space domain of the trimmed surface such that the 3D planar approximation, obtained by mapping 2D triangles onto the surface, deviates from the trimmed surface by no more than [ϵ]. The number of triangles computed in parameter space depends on the bounds of the second derivatives. A detailed discussion of the algorithm and a practical error analysis of the tessellation are provided. Keywords: nurbs; tessellation; visualization © Copyright 2001, Elsevier Science, All rights reserved. ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 27 (2) (1995) pp. 147-157 © 1995 Elsevier Science B.V. All rights reserved. PII: S0010-4485(95)92154-K Application of nurbs in numerical grid generation TzuYi Yu and Bharat K Soni Engineering Research Center, Mississippi State University, PO Box 6176, MS 39762, USA Received 01 June 1994; received 01 September 1994 Abstract The application of nonuniform rational B-splines to grid generation involving complex industrial geometries is discussed. Geometrical entities are transformed and represented usingnurbs with common data structures. Techniques for reparameterization are discussed in terms of various grid topologies and distribution requirements. Computational examples representative of practical computational fluid dynamics applications are presented to demonstrate the success of these methodologies. Keywords: nurbs; grid generation; computational fluid dynamics ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 27 (2) (1995) pp. 139-146 © 1995 Elsevier Science B.V. All rights reserved. PII: S0010-4485(95)92153-J Feature based models for anatomical data fitting Gregory T Dobson, Warren N Waggenspack Jr and Henry J Lamousin Interactive Modeling Research Laboratory, Mechanical Engineering Department, Louisiana State University, Baton Rouge, LA 70803-6413, USA Received 29 May 1994; received 13 September 1994 Abstract Taking advantage of the known, geometric structure of the data being fit, a generic model having this shape is used as an initial approximation. The model is first globally sized and aligned with the data on the basis of the min/max box, principal axes and landmarks. Once aligned, features embedded within the model, corresponding to pertinent characteristics of the data, are adjusted to better fit the data. Finally, vertices and weights of the rational B-spline are optimized to locally deform the model and approximate the data to within a specified tolerance. The feature based modelling approach is demonstrated in 2D by the fitting of facial profiles. Keywords: nurbs; feature based data fitting; facial profiles ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 27 (2) (1995) pp. 129-137 © 1995 Elsevier Science B.V. All rights reserved. PII: S0010-4485(95)92152-I Planar rational B-spline motions Michael G Wagner * Institut für Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/113, A-1040 Wien, Austria Received 26 February 1994; received 09 September 1994 Abstract Nonuniform rational B-spline (nurbs) curves and their associated techniques are of major importance in computer aided geometric design. The paper discusses planar rational B-spline motions. These are planar motions in which all point paths arenurbs curves. Such motions are connected with a linear control structure, which can be used to apply algorithms developed for the design of curves and surfaces directly to the design of planar motions. The first part of the paper gives a brief introduction to plane kinematics and the theory of kinematic mappings. Rational motions and the application of the corresponding control structures are discussed in detail. The second part of the paper presents a C2 interpolation scheme with rational motions of degree 4, which is the minimum degree for motions which have positions with vanishing angular velocity. Keywords: motions; nurbs curves; kinematic mapping ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 27 (2) (1995) pp. 111-127 © 1995 Elsevier Science B.V. All rights reserved. PII: S0010-4485(95)92151-H Dynamic nurbs swung surfaces for physics-based shape design Hong Qin and Demetri Terzopoulos Department of Computer Science, University of Toronto, 10 King's College Road, Toronto, Ontario, M5S 1A4, Canada Received 29 May 1994; received 07 September 1994 Abstract A dynamic, freeform surface model which is useful for representing a broad class of objects with symmetries and topological variability is developed. The new model is based upon swung nurbs surfaces, and it inherits their desirable crosssectional design properties. It melds these geometric features with the demonstrated convenience of surface design within a physics-based framework. Several applications of dynamic nurbs swung surfaces are demonstrated, including interactive sculpting through the imposition of forces, and the adjustment of physical parameters such as mass, damping, and elasticity. Additional applications include surface design with geometric and physical constraints, by rounding solids, and through the fitting of unstructured data. The equations of motion for the dynamic nurbs swung surface model are derived using the Lagrangian mechanics of an elastic surface and the finite element method. It is also shown that these surfaces are a special case of d-nurbs surfaces, a recently proposed physics-based generalization of standard geometric nurbs. The freeform, rational model not only provides a systematic and unified approach for a variety of cagd problems such as constraint-based optimization, variational design, automatic weight selection, and shape approximation, but it also supports interactive sculpting using physics-based manipulation tools. Keywords: nurbs; deformable models; surface blending ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 27 (2) (1995) pp. 95-100 © 1995 Elsevier Science B.V. All rights reserved. PII: S0010-4485(95)92149-M Object-oriented paradigm for nurbs curve and surface design Richard D Fuhra, Lwo Hsiehb and Michael Kallayc a EDS, 13555 SE 36th Street, Bellevue, WA 98006, USAb Wagstaff Inc., N 3910 Flora Road, Spokane, WA 99216, USAc Shapeware, 520 Pike Street, Seattle, WA 98101-4001, USA Received 13 March 1994; received 06 September 1994 Abstract The prevailing paradigm in the design of curve and surface algorithms could be described as being control-points oriented. The limitations of this approach are explored, and an alternative paradigm is presented that combines object-oriented design with two well known approximation algorithms that are rarely mentioned in the cad literature. It provides a powerful and versatile tool for constructing nurbs entities. Keywords: nurbs; curves and surfaces; algorithms ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 27 (2) (1995) pp. 85-93 © 1995 Elsevier Science B.V. All rights reserved. PII: S0010-4485(95)92148-L Unified approach to nurbs curve shape modification C K Au and M M F Yuen Received 27 April 1994; received 01 September 1994 Abstract The shape of nurbs curves can be varied by altering the knot sequences, control point locations and weights of control points. The paper presents a unified approach that modifies the shape of nurbs curves by altering the weights and the location of the control points simultaneously. The effect of the result of the variation is studied for each of the methods. A homogeneous coordinate representation is introduced to provide some physical insight into the weights of the control points. Keywords: nurbs; control points; positional constraints ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 27 (4) (1995) pp. 263-275 © 1995 Elsevier Science B.V. All rights reserved. PII: S0010-4485(95)91136-9 Quasinonparametric surfaces Javier Sánchez-Reyes * Department of Mechanical Engineering, ETSEIB, Polytechnic University of Catalonia, Diagonal 647, 08028 Barcelona, Spain Received 11 September 1993; received 20 November 1993 Abstract The concept of nonparametric surfaces is extended to the definition of quasinonparametric surfaces. These surfaces project one to one onto the plane z = 0 and are built up over the area between two curves that are single-valued, either in Cartesian or in polar coordinates. In the Cartesian case, a subset of nonrational B-spline surfaces is used. The scheme in polar coordinates exhibits all the positive properties of the B-spline tensor-product approach, and yields a piecewise rational Bézier surface. The main interest of this representation is that there exists a very simple point membership classification algorithm for the volume defined between two quasinonparametric patches. Therefore, such a volume can be incorporated as a primitive into a CSG solid-modelling system. Keywords: quasinonparametric patches; nurbs; solid modelling ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 27 (6) (1995) pp. 465-470 © 1995 Elsevier Science B.V. All rights reserved. PII: S0010-4485(95)00019-N Freeform surface region optimization for 3-axis and 5-axis milling Gershon Elber * Department of Computer Science, Technion, Israel Institute of Technology, Haifa 32000, Israel Received 21 March 1994; received 03 June 1994 Abstract A regional optimization of automatically generated machining toolpaths for computer based freeform mechanical models is considered in the framework of multiaxis (3-5) machining operations. A trichotomy of an arbitrary freeform shaped surface into convex, concave and saddle-like regions is considered. Saddle-like and concave regions can be milled using a ball end tool, in 3-axis or 5-axis mode. A subset of the convex regions can be machined faster and with a smaller scallop using a flat end tool that is aligned with the normal of the surface in 5-axis mode. A method to robustly detect and eliminate local gouging of the milling tool into the object is developed and demonstrated on an actual part. Keywords: nurbs; Bézier surfaces; curvature analysis ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 27 (7) (1995) pp. 509-521 © 1995 Elsevier Science B.V. All rights reserved. PII: S0010-4485(94)00003-V Efficient rendering of trimmed NURBS surfaces Subodh Kumar and Dinesh Manocha Department of Computer Science, University of North Carolina, Chapel Hill, NC 27599, USA Received 02 June 1994 Abstract An algorithm for the interactive display of trimmed nurbs surfaces is presented. The algorithm converts the nurbs surfaces to Bézier surfaces, and nurbs trimming curves to Bézier curves. It tessellates each trimmed Bézier surface into triangles, and renders them using the triangle rendering capabilities common in current graphics systems. It makes use of tight bounds for the uniform tessellation of Bézier surfaces into cells and it traces the trimming curves to compute the trimmed regions of each cell. This operation is based on the tracing of trimming curves, intersection computation with the cells, and triangulation of the cells. The resulting technique also makes use of spatial and temporal coherence between successive frames for cell computation and triangulation. Polygonization anomalies such as cracks and angularities are avoided as well. The algorithm can display trimmed models described using thousands of Bézier surfaces at interactive frame rates on high end graphics systems. Keywords: trimmed nurbs; Bézier surfaces; rendering ;------------------------------------------------------------------------------- Computer-aided Design, Vol. 27 (2) (1995) pp. 101-110 © 1995 Elsevier Science B.V. All rights reserved. PII: S0010-4485(95)92150-Q Algorithm for degree reduction of B-spline curves Les Piegla and Wayne Tillerb a Department of Computer Science & Engineering, University of South Florida, 4202 Fowler Avenue, ENG 118, Tampa, FL 33620, USAb GeomWare Inc., 3036 Ridgetop Road, Tyler, TX 75703-8906, USA Received 29 March 1994; received 01 September 1994 Abstract An algorithmic approach to degree reduction of B-spline curves is presented. The method consists of the following steps: (a) decompose the B-spline curve into Bézier pieces on the fly, (b) degree reduce each Bézier piece, and (c) remove the unnecessary knots. A complete algorithm and precise error control are provided. Keywords: B-splines; degree reduction; curves and surfaces ;------------------------------------------------------------------------------- [CRES97c] Distance between a point and a convex hull : proofs of two proximity lemmas. B. Crespin Rapport technique n° RR_ 118997, LaBRI, Université Bordeaux I, 1997. ;------------------------------------------------------------------------------- [GRIS97a] L'approche hermitienne pour les splines ŕ multi-résolution. L. Grisoni, C. Schlick In : Actes des Journées AFIG 97, pp 247-256, (Rennes), Décembre 1997. ;------------------------------------------------------------------------------- [GRIS97b] An hermitian approach for multiresolution splines. L. Grisoni, C. Blanc, C. Schlick Rapport technique n° 119297LaBRI, Université Bordeaux I, 1997. ;------------------------------------------------------------------------------- [GRIS97c] HB-splines : an interesting subset of B-splines for interactive CAD applications. L. Grisoni, C. Blanc, C. Schlick Rapport technique n° 119397LaBRI, Université Bordeaux I, 1997. ;------------------------------------------------------------------------------- ;------------------------------------------------------------------------------- Approximating NURBS Curves by Arc Splines Xunnian Yang Zhejiang University It is desirable to approximate a smooth curve by arc spline with fewest segments within prescribed tolerance. In this paper, we present an efficient algorithm for fitting planar smooth curve by arc spline. The main idea is that we construct the optimal arc spline by optimizing the interpolating biarc curve. The scheme consists of three steps: (1) sampling the curve based on consecutive tangents deviation; (2) construct the interpolating arc spline; (3) reduce the arc number to the minimum within prescribed tolerance. The algorithm can control the approximating error efficiently and results in the fewest number of arc segments. Keywords: smooth curve, arc spline, NC tool path generation, data reduction Proceedings of the Geometric Modeling and Processing 2000 Copyright (c) 2000 Institute of Electrical and Electronics Engineers, Inc. All rights reserved. ;------------------------------------------------------------------------------- Constructing NURBS Surface Model from Scattered and Unorganized Range Data In Kyu Park and Sang Uk Lee Seoul National University Il Dong Yun Hankuk University In this paper, we propose an algorithm to produce 3-D surface model from a set of range data, based on Non-Uniform Rational B-Splines (NURBS) surface fitting technique. It is assumed that the range data is initially unorganized and scattered 3-D points, while their connectivity is also unknown. The proposed algorithm is roughly made up of two stages: initial model approximation employing K-means clustering and construction of NURBS patch network using hierarchical graph representation. The initial model is approximated by both polyhedral and triangular model. Then, the initial model is represented by hierarchical graph, which is efficiently used to construct the G 1 continuous NURBS patch network of the whole object. Experiments are carried out on synthetic and real range data to evaluate the performance of the proposed algorithm. It is shown that the initial model as well as the NURBS patch network are constructed automatically, while the modeling error is observed to be negligible. Proceedings of the Second International Conference on 3-D Imaging and Modeling Copyright (c) 1998 Institute of Electrical and Electronics Engineers, Inc. All rights reserved. ;------------------------------------------------------------------------------- A Shape-Preserving Data Embedding Algorithm for NURBS Curves and Surfaces Ryutarou Ohbuchi Yamanashi University Hiroshi Masuda The University of Tokyo Masaki Aono IBM Existing data embedding algorithms for polygonal meshes and their attributes can't be applied to the majority of (geometric) computer aided design (CAD) applications, for two major reasons. First, these CAD systems employ parametric curves and surfaces, not polygonal meshes, as their main shape-defining primitives. Second, most CAD applications do not tolerate modifications of model topology and/or geometry that are introduced by existing data embedding algorithms. This paper proposes a new data embedding algorithm for non-uniform rational B-spline (NURBS) curves and surfaces, which employs rational linear reparameterization for embedding messages. The algorithm exactly preserves the shape, - that is, the geometry and topology - of its embedding targets. Furthermore, it preserves the data size of the model. We consider these two properties, preservation of shape and preservation of data size, can be significant with regard to the use of data embedding in CAD applications. In addition to the shape- and data size-preserving data embedding algorithm for NURBS curves and surfaces, this paper outlines additional methods for embedding data in various types of parametric curves and surfaces. Keywords: computer-aided design (CAD), information security, digital watermark, reparameterization. Proceedings of the Computer Graphics International Copyright (c) 1998 Institute of Electrical and Electronics Engineers, Inc. All rights reserved. ;------------------------------------------------------------------------------- Interactive Display of Large NURBS Models Subodh Kumar, Dinesh Manocha, and Anselmo Lastra Abstract: We present algorithms for interactive rendering of large-scale NURBS models. The algorithms convert the NURBS surfaces to Bézier surfaces, tessellate each Bézier surface into triangles, and render them using the triangle-rendering capabilities common to current graphics systems. This paper presents algorithms for computing tight bounds on surface properties in order to generate high quality tessellation of Bézier surfaces. We introduce enhanced visibility determination techniques and present methods to make efficient use of coherence between successive frames. In addition, we also discuss issues in parallelization of these techniques. The algorithm also avoids polygonization anomalies like cracks. Our algorithms work well in practice and, on high-end graphics systems, are able to display models described using thousands of Bézier surfaces at interactive frame rates. Index Terms: NURBS, tessellation, triangulation, visibility, interactive display, CAD, parallel algorithm. S. Kumar is with the Department of Computer Science, Johns Hopkins University, Baltimore, MD 21218-2694. E-mail: subodh@cs.jhu.edu. D. Manocha and A. Lastra are with the Department of Computer Science, University of North Carolina, Chapel Hill, NC 27599. E-mail: {manocha, lastra}@cs.unc.edu. IEEE Transactions on Visualization and Computer Graphics, Vol. 2, No. 4, December 1996 Copyright (c) 1996 Institute of Electrical and Electronics Engineers, Inc. All rights reserved. ;------------------------------------------------------------------------------- A High Precision 3D Object Reconstruction Method Using a Color Coded Grid and NURBS Philippe Lavoie, Dan Ionescu and Emil Petriu University of Ottawa A high precision technique for reconstructing three dimensional (3D) objects from its one-camera 2D views and using a color coded structured light is described. The method does not require any a-priori knowledge of the absolute positioning and orientation of the camera, nor of a point in the scene. However, a calibration process precedes the 3D reconstruction algorithm leading to a high precision calculation of the position of the camera. The structured light, which is pseudo-randomly color encoded, is used to determine unique points and lines on the surface of the object. The lines once extracted from a view of the object, are used for the determination of non-uniform rational Bézier splines (NURBS) curves to generate the surface of the object. The precision of the technique depends on the adaptation of the sampling rate of the structured light to the surface of the object. In a recent test, the precision of 7 tau (thousands of an inch) was achieved with a relatively smooth object. The paper describes only the 3D reconstruction process and not the integration of multiple views into one single 3D image of the object. Proceedings of the 10th International Conference on Image Analysis and Processing Copyright (c) 1998 Institute of Electrical and Electronics Engineers, Inc. All rights reserved. ;------------------------------------------------------------------------------- Simple Technique for NURBS Shape Modification Javier Sanchez-Reyes Polytechnic University of Catalonia Abstract: This article presents a unified approach to NURBS shape modification, based on a perspective functional transformation of arbitrary origin. This procedure alters simultaneously the weight and location of a control point, and encompasses as particular cases the standard displacement of a control point or modification of a weight. Thus, it provides a homogeneous interface for NURBS shape modification, based on simple, easily understandable geometric concepts. This method works not only for curves, but also for NURBS surfaces or volumes. In the important case of rational quadratic Bezier curves (that is, conics), this shape handling is related to the classic concepts of focus and eccentricity. Finally, I develop applications for constrained NURBS manipulation. Keywords: NURBS, perspective transformation, shape, conics, constrained manipulation IEEE Computer Graphics and Applications, Vol. 17, No. 1, January - February 1997 Copyright (c) 1997 Institute of Electrical and Electronics Engineers, Inc. All rights reserved. ;------------------------------------------------------------------------------- Integrated Computer Graphics Learning System in Virtual Environment - Case Study of Bézier, B-Spline, and NURBS Algorithms Wen-Chai Song, Shih-Ching Ou and Song-Rong Shiau National Central University The Web is a popular platform for training educators and learners in educational systems. We find several Web-based training systems that are developed by enterprises that try to train teachers and students to use Web technologies in educational settings. The rapid growth of the Internet and computer technology in recent years has sparked a demand for easily accessible and affordable learning solutions in corporate offices, classrooms, and homes. Traditional learning models being challenged by the fast-paced, real-time environments available today. The purpose of this paper is to apply systems design principles to the design of Web-based training systems for teachers and learners. Several issues will be addressed in this paper such as the development of an Integrated Learning Graphics System (ILGS) for a better learning environment. In this paper, we attempted to develop a web-based learning system of computer graphics by Bézier, B-spline and NURBS algorithms. The purpose of the research was to increase the effect of Computer-Aided Design (CAD) in network .The other advantages is that network browser is the common platform in internet and intranet, the graphics system can be portable cross different operating system, as like windows 95,linux. In fact, the graphics system was an attempt to be shared the resource each other. Keywords: Distance Learning, Virtual Reality, Web-Based, NURBS Proceedings of the IEEE International Conference on Information Visualization (IV2000) Copyright (c) 2000 Institute of Electrical and Electronics Engineers, Inc. All rights reserved. ;------------------------------------------------------------------------------- Representing Spheres and Ellipsoids Using Periodic NURBS Surfaces with Fewer Control Vertices Kaihuai Qin and Zesheng Tang Tsinghua University Wenping Wang The University of Hong Kong A new kind of NURBS representation for periodic curves is introduced. The method for periodic curves is extended to derive a compact and symmetric representation of the sphere or the ellipsoid using a piecewise bi-quadratic NURBS surface with only eight distinct control vertices. Proceedings of the Pacific Graphics Copyright (c) 1998 Institute of Electrical and Electronics Engineers, Inc. All rights reserved. ;------------------------------------------------------------------------------- ;------------------------------------------------------------------------------- ;------------------------------------------------------------------------------- ;------------------------------------------------------------------------------- ;------------------------------------------------------------------------------- ;------------------------------------------------------------------------------- ;------------------------------------------------------------------------------- ;------------------------------------------------------------------------------- ;-------------------------------------------------------------------------------