Bicubic Bezier Research Articles

Содержание курса «Геометрическое моделирование» для направления подготовки «Математика и компьютерные науки»

In this paper has been considered the main content and distinctive features of the “Geometric Modeling” training course for the “Mathematics and Computer Science” training program 02.03.01 (“Mathematical and Computer Modeling” specialization).
 The goal of the “Geometric Modeling” course study is the assimilation of mathematical methods for construction of geometric objects with complex curved shapes, and techniques for their computer visualization by using polygons of curves and surfaces. Methods for construction of structures’ curved shapes using spline representations, as well as techniques for construction of surfaces and volumetric geometries using motion operations and basic logical operations on geometric objects are considered. The spline representations include linear and bilinear splines, Hermite cubic splines and Hermite surfaces, natural cubic and bicubic interpolation splines, Bezier curves and surfaces, rational Bezier splines, B-splines and B-spline surfaces, NURBS-curves and NURBS-surfaces, transfinite interpolation methods, and splines of surfaces with triangular form. Logical operations for intersection of two spline curves, and intersection of two parametric surfaces are considered. The principles of scientific visualization and computer animation are considered in this course as well.
 Some examples for visualization of initial data and results of curves and surfaces construction in two- and three-dimensional spaces through the software shell developed by authors and used by students while doing tests have been demonstrated. The software shell has a web interface with the WebGL library graphic support. Tasks for four practical studies in a computer classroom, as well as several variations of homework are represented.
 The problems occurring in preparation materials for some course sections are discussed, as well as the practical importance of acquired knowledge for the further progress of students.
 The paper may be interesting for teachers of “Geometric Modeling” and “Computer Graphics” courses aimed to students with a specialization in mathematics and information, as well as to those who independently develop software interfaces for algorithms of geometric modeling.

A generalised formulation of G-continuous Bezier elements applied to non-linear MHD simulations

The international tokamak ITER is progressing towards assembly completion and first-plasma operation, which will be a physics and engineering challenge for the fusion community. In the preparation for ITER experimental scenarios, non-linear MHD simulations are playing an essential role to actively understand and predict the behaviour and stability of tokamak plasmas in future fusion power plant. The development of MHD codes like JOREK is a key aspect of this research effort, and provides invaluable insight into the plasma stability and the control of global and localised plasma events, like Edge-Localised-Mode and disruptions. In this paper, we present an operational implementation of a new, generalised formulation of Bezier finite-elements applied to the JOREK code, a significant advancement from the previously G1-continuous bi-cubic Bezier elements. This new mathematical method enables any polynomial order of Bezier elements, with a guarantee of G-continuity at the level of (n−1)/2, for any odd n, where n is the order of the Bezier polynomials. The generalised method is defined, and a rigorous mathematical proof is provided for the G-continuity requirement. Key details on the code implementation are mentioned, together with a suite of tests to demonstrate the mathematical reliability of the finite-element method, as well as the practical usability for typical non-linear tokamak MHD simulations. A demonstration for a state-of-the-art simulation of an Edge-Localised-Mode instability in the future ITER tokamak, with realistic grid geometry, finalises the study.

An efficient multipoint 5-axis tool positioning method for tensor product surfaces

In this work, a robust tool-drop algorithm coupled with the bisection method is used to speed up the computations in the drop and tilt method (DTM), a multipoint tool positioning technique for 5-axis machining with a toroidal end mill. In the DTM, the tool is dropped on the surface along a predetermined tool-axis direction until it touches the surface tangentially. The position of the circular tool insert (pseudo-insert circle) through the first point of contact is then used to compute the tool-tilt angle that causes the tool to touch the surface tangentially at a second point. A previous implementation of DTM used Newton’s method to solve the system of equations for tilting the tool, but took many iterations and multiple initial seed solutions to converge to a valid solution for tool-tilt orientation. Using the bisection method coupled with the generalized tool-drop algorithm instead of a direct use of Newton’s method to determine the tool-tilt angle improves performance without compromising the accuracy. The method was implemented in C+ +, and a comparative study with an earlier method for several bi-quadratic and bi-cubic Bezier surfaces demonstrates the reliability of the developed method.

On G1 stitched bi-cubic Bézier patches with arbitrary topology

• Lower bounds on G 1 bi-cubic surfaces imply restrictions on layout. • G 1 stitched bi-cubic Bezier patches with arbitrary topology require multiple pieces. • Pre-refinement by Doo –Sabin subdivision as in [1] does not guarantee mathematical smoothness.

Toward design optimization of a Pelton turbine runner

The objective of the present paper is to propose a strategy to optimize the performance of a Pelton runner based on a parametric model of the bucket geometry, massive particle based numerical simulations and advanced optimization strategies to reduce the dimension of the design problem. The parametric model of the Pelton bucket is based on four bicubic Bezier patches and the number of free parameters is reduced to 21. The numerical simulations are performed using the finite volume particle method, which benefits from a conservative, consistent, arbitrary Lagrangian Eulerian formulation. The resulting design problem is of High-dimension with Expensive Black-box (HEB) performance function. In order to tackle the HEB problem, a preliminary exploration is performed using 2'000 sampled runners geometry provided by a Halton sequence. A cubic multivariate adaptive regression spline surrogate model is built according to the simulated performance of these runners. Moreover, an original clustering approach is proposed to decompose the design problem into four sub-problems of smaller dimensions that can be addressed with more conventional optimization techniques.

Intersection between Bi-cubic Bezier Curved Surface and Plane

The paper presents a solution to calculate intersection between a plane and bi-cubic Bezier curved surface. The paper first constructs bounding box based on the control grid by the characteristics of the Bezier surface, and then makes use of the plane half-space properties to convert the problem that calculates the intersection line between the plane and the parameter curved surface into calculating a series of intersection points by using the method that apply the bi-linear interpolation algorithm to the curved surface in representation of parameter area. The intersection line between the plane and the surface can be calculated by connecting the intersection points. This method can solve the surface intersection line well whether the line is discontinuous or form a closed cycle. The algorithm is relatively simple, and has good adaptability for calculating the intersection line between the plane and the curved surface. urround��dr����I�o-page-border-surround-footer:no;} @page Section1 {size:612.0pt 792.0pt; margin:72.0pt 90.0pt 72.0pt 90.0pt; mso-header-margin:36.0pt; mso-footer-margin:36.0pt; mso-paper-source:0;} div.Section1 {page:Section1;} --> Multiresolution technique is one of the most efficient approaches to improve the rendering performance, but its design and implementation for massive meshes are still very difficult. This paper researches and realizes constructing and rendering of multiresolution representation for massive meshes base on surface partition, which could provide vertex-grained local refinement and generate the optimal rendering quality. Our approach adopts dual hierarchy to represent the mesh. One is cluster hierarchy of progressive meshes for coarse-grained selective refinement. The other is vertex hierarchy built with progressive mesh in the cluster node to provide fine-grained local refinement. In order to promote the speed of local refinement, we introduce some data structures and dependency rules to realize parallel view-dependent refinement for vertex hierarchy by using GPU, which greatly reduces the load of CPU and enables it to prefetch data to hide I/O latency effectively. In addition, we propose a new mesh layout algorithm which reorders triangles contained by cluster node to reduce the average cache miss ratio and further improve the rendering speed.

Fuzzy set theory in modeling uncertainty data via interpolation rational Bezier surface function

Lately, the research of rational spline function widely expanding because the designing of curve and surface, which is the additional parameter known as weight that it had. This weight use to changing the curve and surface shape to fit the real data form. Then, the interpolation method adapted in rational spline function as interpolation rational spline function, which can be constructed by using piecewise rational spline function. This method is being used to model the collection of data set to visualize the data for more the understanding the meaning of the data through the curve especially surface forms. However, the problem arises when the data become uncertain, which is unable to model them. Therefore, in this paper, we introduced fuzzy interpolation rational bicubic Bezier surface (later known as FIRBBS) which modeling the fuzzy data after had been defined by the fuzzy sets definition. The construction of FIRBBS is based on the definition of fuzzy set theory and interpolation rational bicubic Bezier surface. Then, the illustration given in figures for more understanding the method.

3D Reconstruction Using Cubic Bezier Spline Curves and Active Contours (Case Study)

Introduction 3D reconstruction of an object from its 2D cross-sections (slices) has many applications in different fields of sciences such as medical physics and biomedical engineering. In order to perform 3D reconstruction, at first, desired boundaries at each slice are detected and then using a correspondence between points of successive slices surface of desired object is reconstructed. Materials and Methods In this study, Gradient Vector Flow (GVF) was used in order to trace the boundaries at each slice. Then, cubic Bezier Spline curves were used to approximate each of obtained contours and to approximate the corresponding points of different contours at successive slices. The reconstructed surface was a bi-cubic Bezier Spline surface which was smooth with G2 continuity. Results Our presented method was tested on SPECT data of JASZCZAK phantom and human's left ventricle. The results confirmed that the presented method was accurate, promising, applicable, and effective. Conclusion Using GVF algorithm to trace boundaries at each slice, and cubic Bezier Spline curves to approximate the obtained rough contours yield to the procedure of reconstruction which was fast and also the final surface was smooth with G2 continuity. So far, some mathematical curves such as spline, cubic spline, and B-spline curves were used to approximate the computed contour during a time consuming procedure. This study presented a 3D reconstruction method based on a combination of GVF algorithm and cubic Bezier Spline curves. There was a good trade-off between speed and accuracy in using cubic Bezier Spline curves which is especially useful for training students.

N-way blending problem of circular quadrics

A method of blending circular quadrics with parametric patches is proposed in this paper. It needs n rational bi-cubic Bezier patches and two S-patches to blend n (n > 2) quadrics. Bi-cubic rational Bezier patches are used to G1-continuously blend each pair of adjacent base surfaces. These patches enclose two n-sided holes. And an S-patch is used to fill each hole, which keeps nearly G1-continuity. Explicit formulas of control points for both Bezier patches and S-patches are derived from their constraint conditions, which make modeling by blending quadrics very simple and convenient. In addition, the shapes of the blending surfaces can be intuitively adjusted by free parameters.

Three dimensional modeling and finite element simulation of a generic end mill

The geometry of cutting flutes and the surfaces of end mills is one of the crucial parameters affecting the quality of the machining in the case of end milling. These are usually represented by two-dimensional models. This paper describes in detail the methodology to model the geometry of a flat end mill in terms of three-dimensional parameters. The geometric definition of the end mill is developed in terms of surface patches; flutes as helicoidal surfaces, the shank as a surface of revolution and the blending surfaces as bicubic Bezier and biparametric sweep surfaces. The proposed model defines the end mill in terms of three-dimensional rotational angles rather than the conventional two dimensional angles. To validate the methodology, the flat end milling cutter is directly rendered in OpenGL environment in terms of three-dimensional parameters. Further, an interface is developed that directly pulls the proposed three-dimensional model defined with the help of parametric equations into a commercial CAD modeling environment. This facilitates a wide range of downstream technological applications. The modeled tool is used for finite element simulations to study the cutting flutes under static and transient dynamic load conditions. The results of stress distribution (von mises stress), translational displacement and deformation are presented for static and transient dynamic analysis for the end mill cutter flute and its body. The method described in this paper offers a simple and intuitive way of generating high-quality end mill models for use in machining process simulations. It can be easily extended to generate other tools without relying on analytical or numerical formulations.

Modeling of twist drills in terms of 3D angles

Traditional nomenclatures of specifying cutting tool geometries are two-dimensional (2D) in nature. The present work presents a paradigm to model the geometries of a variety of twist drills in terms of three-dimensional (3D) parameters. The work outlines the construction of a detailed computer-aided design (CAD) model for a fluted twist drill and establishes a new 3D definition for the geometry of drill in terms of biparametric surface patches. The flutes of the drill are modeled as helicoidal surfaces. For this, sectional geometry of tip-to-tip profile is developed and then swept. The geometric model of the shank is developed separately. The transitional surfaces are modeled as bicubic Bezier surfaces. With this methodology, we propose a new 3D nomenclature for drill geometries in terms of 3D rotational angles. The relations necessary to map the proposed three-dimensional angles to two-dimensional conventional angles, known as forward mapping and their reverse relations (inverse mapping) are also developed. The new paradigm offers immense technological advantages in terms of numerous downstream applications.

A new class of term orders for elimination

Elimination is a classical subject. The problem is algorithmically solvable by using resultants or by one calculation of Groebner basis with respect to an elimination term order. However, there is no existing method that is both efficient and reliable enough for applicable size problems, say implicitization of bi-cubic Bezier surfaces with degree six in five variables. This basic and useful operation in computer aided geometric design and geometric modeling defies a solution even when approximation using floating-point or modular coefficients is used for Groebner basis computation. An elimination term order can be used to eliminate U for any ideal in K [ X ] [ U ] . However, for most practical problems we are given a fixed ideal, which means that an elimination term order may be too much for our calculation. In this paper, the author proposes a new approach for elimination. Instead of using a classical elimination term order for all problems or ideals as usual, the author proposes to use algebraic structures of the given system of equations for finding more suitable term orders for elimination of the given problem only. Experimental results showed that these ideal-specific term orders are much more efficient for elimination. In particular, when ideal specific term orders for elimination are used with Groebner walk conversion, one can completely avoid all perturbations. This is a significant result because researchers have been struggling with how to perturb basis conversions for a long time.

Adaptive triangular element generation and optimization-based smoothing: Part 2. On the surface

A general and systematic approach to adaptive triangular element generation and optimization-based smoothing on the surface is presented in this paper. The approach starts from a rough triangular mesh that needs to be remeshed in order to meet the desired mesh density. Various local transformation processes including edge and face splitting, edge collapsing and edge swapping are employed for enhancement of mesh quality as well as for generation of new triangular elements from the rough triangular elements. An edge swapping scheme is proposed, which satisfy both of the Delaunay criterion and topological consistency. To reduce the geometrical error that is inevitable in remeshing, curvature of the surface is not only considered in determining the mesh density distribution but the characteristic edges and the triangular elements are also interpolated by the third-order chord-length spline curves and the bi-cubic Bezier patches, respectively. The nodal points of the generated mesh are smoothed by the physically-based smoothing scheme and then optimally placed by the optimization-based smoothing scheme. Through various application examples, the characteristics and effects of the presented approaches are quantitatively investigated.

Reconstruction from contour lines based on bi-cubic Bézier spline surface

A novel reconstruction method from contours lines is provided. First, we use a simple method to get rid of redundant points on every contour, then we interpolate them by using cubic Bezier spline curve. For corresponding points of different contours, we interpolate them by the cubic Bezier spline curve too, so the whole surface can be reconstructed by the bi-cubic Bezier spline surface. The reconstructed surface is smooth because every Bezier surface is patched with G2 continuity, the reconstruction speed is fast because we can use the forward elimination and backward substitution method to solve the system of tridiagonal equations. We give some reconstruction examples at the end of this paper. Experiments showed that our method is applicable and effective.

On the B-spline interpolated tool trajectories for five-axis sculptured surface machining

B-spline interpolation scheme is now available on modern five-axis Computer Numerical Control (CNC) machine tools. With this newly implemented interpolation scheme, a cutting tool can be directly commanded to trace B-spline trajectories, which approximate ideal 3D curved trajectories, in sculptured surface machining. The approximation of ideal tool trajectories by B-spline interpolated tool trajectories inevitably leads to machining errors, referred to as the geometry-based errors in the present work. It is essential to ensure synchronisation of the movements of the three translational and two rotational joints of a five-axis machine tool to reduce the geometry-based errors. This paper presents an effective method to achieve synchronisation of the machine joint movements. It first fits a 3D B-spline for the three translational joints and then uses a knot inheriting procedure to fit a 2D B-spline for the two rotational joints. Evaluation of the presented method was made through the machining of a typical bi-cubic Bezier surface on a five-axis machine tool capable of performing non-uniform B-spline interpolation. It was found that the resulting geometry-based errors, which were varying along the given isoparametric tool paths, were able to be maintained below 25m.

Distortion Theory of Stereographic Draping with Flexible Fabric on General Surface: Part I-Geometric Analysis

Stereographic Draping is a mathematical model of garment draping that unfolds a 3-D garment surface to the corresponding flat garment patterns. This unfolding process is accurate only when the fabric is rigid. In general, it induces distortion in distance and area. The distortion is related to the intrinsic properties of the garment surface and the mechanical properties of the fabric. Mathematically, Stereographic Draping on a general surface is very difficult to be analyzed. Based on representation of the bicubic Bezier patch, the theoretical analysis was conducted to formulate the distortion properties of Stereographic Draping. This article is the part one of the results of the theoretical analysis. The distortion is expressed in terms of the intrinsic geometric properties of the garment surface. In part two, the focus will be on the distortion properties with respect to the fabric mechanical properties. The distortion theory provides insight in the garment pattern design theory and addresses the distortion properties.

Determination of the Optimal Parameters for Freeform Surface Measurement and Data Processing in Reverse Engineering

One objective of this work is to determine the optimal combination of the probe diameter and grid distance for freeform surface measurement, and another is to determine the optimal parameters for the local Shepard interpolation. The optimal combination of the probe diameter and grid distance for freeform surface measurement was determined through a Taguchi matrix experiment. The smaller the probe diameter and grid distance, the better the accuracy of the surface normal based on the configured matrix experimental result. The optimal parameters, namely the exponent μ and the radius R, for the local Shepard interpolation were determined by using the minimisation method of the root-mean-square normalised error (RMSNE) between the measured data points and the theoretical data points on a standard steel ball surface. The optimal parameters determined were actually applied to the measurement of a freeform surface (mouse surface) on a coordinate measuring machine (CMM). The local Shepard interpolation method was used to interpolate 16 control points from 1054 measured data points. Bi-cubic Bezier- and B-spline surface CAD models were constructed through these interpolated control points.

Optimum design of composite structures with curved fiber courses

In this study, a new methodology for the optimum design of laminated composites with curved fiber courses is presented. The objective of the optimization problem is to minimize the weight of the composite laminate under stress constraints. The Tsai–Hill criterion is employed on the first ply failure basis. Layer thicknesses and fiber angles are represented by bicubic Bezier surfaces and cubic Bezier curves, respectively. Design variables are coordinates of control points of the corresponding Bezier splines. The design variable linking procedure is used in order to reduce the number of design variables further and to provide symmetry in the design. The sequential quadratic programming is used in the optimization. The modeling of the laminate is carried out by using three-node shell finite elements.

Calculation of the Normal Vector using the 3 × 3 Moving Mask Method for Freeform Surface Measurement and its Application

The concept of a 3 3 3 moving mask operation image-processing technique is proposed to calculate the normal vector of measuring points in this study. The method developed reduces greatly the calculation time of matrix operation and memory space in comparison with the traditional composite Ferguson-spline method. The methodology for calculating the normal vector is to select eight neighbouring points at equal distances in the vicinity of an arbitrary node on the surface, from which a small surface patch can then be constructed from the nine selected points. Different analytical methods are used to calculate the unit normal vector, namely the Bezier method with uniform parameters and the Bezier method with non-uniform parameters, and are discussed in this study. The accuracy of these two methods in calculating the unit normal vector was also verified by calculating different positions on a spherical surface. The Shepard interpolation method was adopted to interpolate a few control points from a massive number of measured data points to establish the CAD model of a freeform surface using a rectangular grid. The method developed was applied for the measurement of a freeform surface (mouse surface) using a coordinate measuring machine. The local Shepard interpolation method was used to interpolate 16 control points from 1054 measured data points. A bi-cubic Bezier- and B-spline surface CAD model were constructed through these interpolated control pointer.

Precise Fitting of Rational Bezier Surfaces onto Free Surfaces.

For computer-aided geometric design or for computer-aided geometric display, parametric curve segments and surface patches such as the Bezier form or B-spline form have been frequently used. It is very convenient if we can use the same parametric representations for the approximation of algebraic curves or surfaces, such as circles, ellipses, spheres and tori. Then we can perform unified treatment for all free-form and nonfree-form curves and surfaces. For the design, machining and inspection of accurate geometry such as precision machine parts, it is very important to know the qualitative and quantitative error of approximation from exact geometry. But up to the present, little work has been done on this error estimation. In this report, we treated the method of optimum fitting of rational cubic Bezier curves and rational bicubic Bezier surfaces onto algebraic and free curves and surfaces, and numerically examined the magnitudes of approximation errors.