Approximate merging of multiple Bézier segments

This paper deals with the approximate merging problem of multiple adjacent Bézier curves with different degrees by a single Bézier curve, which is a frequently seen problem in modeling. The unified matrix representation for precise merging is presented and the approximate merging curve is further derived based on matrix operation. Continuity at the endpoints of curves is also discussed in the merging process. Examples show that the method in this paper achieves satisfying merging results.     

Constrained Bézier curves'' best multi-degree reduction in the L2-norm

In computer aided geometric design, the degree reduction of the parameter curve is a key technique in data exchange and data compression. The various existing methods of degree reduction cannot decide whether the degree reduction curve satisfying the given tolerance exists beforehand, cannot give approximation of the best multi-degree reduction, or cannot provide explicit expression and error formula of the degree reduction curve. In this paper, we propose an entirely new method, which can hurdle the above flaws completely. For a given Bézier curve of degree n, we can easily decide whether a Bézier reduction curve of degree m exists, which has equal derivatives with the given curve up to (r-1)-th and (s-1)-th orders (r,s≤m) respectively at the endpoints, so the approximating error is less than the given tolerance ε in the L2-norm. If the curve exists, the explicit expression can be given.     

Curve interpolation based on Catmull-Clark subdivision scheme

An efficient algorithm for curve interpolation is proposed. The algorithm can produce a subdivision surface that can interpolate the predefined cubic B-spline curves by applying the Catmull-Clark scheme to a polygonal mesh containing "symmetric zonal meshes", which possesses some special properties. Many kinds of curve interpolation problems can be dealt with by this algorithm, such as interpolating single open curve or closed curve, a mesh of nonintersecting or intersecting curve. The interpolating surface is C2 everywhere excepting at a finite number of points. At the same time, sharp creases can also be modeled on the limit subdivision surface by duplicating the vertices of the tagged edges of initial mesh, i.e. the surface is only C0 along the cubic B-spline curve that is defined by the tagged edges. Because of being simple and easy to implement, this method can be used for product shape design and graphic software development.     

New algorithms for evaluating parametric surface

Through generalization of mathematical model of surface lofting program in the CONSURF system, the definitions for two generalized Ball surfaces and their recursive algorithms are given. Furthermore, the conversion al gorithms from Bézier surface to these two generalized Ball surfaces are presented. On the basis of these algorithms, two more efficient algorithms for evaluating parametric surfaces are also derived. One uses generalized Ball forms directly for evaluating surface, and the other converts the given Bézier surface to a generalized Ball surface firstly, and then evalu ates the surface. Both theoretical analysis and example computations show that the two new algorithms are more efficient than the de Casteljau algorithm. Especially when Wang-Ball surface is used, the time complexity is reduced from cubic to quadratic of the degree of the surface. If these algorithms are applied to displaying, interactive rendering, designing, intersection-finding, offsetting and approximating for surfaces, considerable economic results can be achieved.     

A new method for automatically constructing convexity-preserving interpolatory splines

Constructing a convexity-preserving interpolating curve according to the given planar data points is a problem to be solved in computer aided geometric design (CAGD). So far, almost all methods must solve a system of equations or recur to a complicated iterative process, and most of them can only generate some function-form convexity-preserving interpolating curves which are unaccommodated with the parametric curves, commonly used in CAGD systems. In order to overcome these drawbacks, this paper proposes a new method that can automatically generate some parametric convexity-preserving polynomial interpolating curves but dispensing with solving any system of equations or going at any iterative computation. The main idea is to construct a family of interpolating spline curves first with the shape parameter a as its family parameter; then, using the positive conditions of Bernstein polynomial to respectively find a range in which the shape parameter a takes its value for two cases of global convex data points and piecewise convex data points so as to make the corresponding interpolating curves convexity-preserving and C2(or G1) continuous. The method is simple and convenient, and the resulting interpolating curves possess smooth distribution of curvature. Numerical examples illustrate the correctness and the validity of theoretical reasoning.     

Improved Ternary Subdivision Interpolation Scheme

An improved ternary subdivision interpolation scheme was developed for computer graphics applications that can manipulate open control polygons unlike the previous ternary scheme, with the resulting curve proved to be still C2-continuous. Parameterizations of the limit curve near the two endpoints are given with expressions for the boundary derivatives. The split joint problem is handled with the interpolating ternary subdivision scheme. The improved scheme can be used for modeling interpolation curves in computer aided geometric design systems, and provides a method for joining two limit curves of interpolating ternary subdivisions.     

Integral computation relating to rational curves based on approximate degree reduction

A new algorithm is presented for computing the volume of revolution, moment of area and centroid etc. , which are related to the integration of rational curves. In this algorithm rational curve with high degree is firstly approximated by those with lower degree through endpoint interpolation. And finally the closed form integration solution is derived for quadratic rational curve. The diminishing rate of the minimum norm of the perturbation vector needed by degree reduction is 0(2" ") when the interval is subdivided at the midpoint. Combining the subdivision with the degree reduction, we can obtain a faster convergence of integration approximation. A series of integral error bound functions which are fairly simple to compute are derived. The examples given in this paper show that this algorithm is a simple and time-saving method in computing with small tolerance.     

A New Algorithm for Feature Matching in Reverse Engineering

Feature recognition and surface reconstruction from point clouds are difficulties in reverse engineering. A new surface reconstruction algorithm for slicing point cloud was presented. The contours of slice were extracted. Then, the intersection of two adjacent curve segments in the contour was obtained and curves feature was extracted. Finally, adjacent section contours were matched directly with Fourier-Mellin curve matching method for feature extraction. An example of 3-D model reconstruction shows the reliability and application of the algorithm.     

Elliptic curves in Huff’s model

In this paper,we present the generalized Huff curves that contain Huff’s model as a special case.First,it is proved that every elliptic curve with three points of order 2 is isomorphic to a generalized Huff curve.Then,the fast and explicit formulae are derived for generalized Huff curves in projective coordinates.This paper also enumerates the number of isomorphism classes of generalized Huff curves over finite fields.Finally,the explicit formulae are presented for the doubling step and addition step in Miller’s algorithm to compute the Tate pairing on generalized Huff elliptic curves.     

Planar cubic hybrid hyperbolic polynomial curve and its shape classification

The planar cubic hybrid hyperbolic polynomial curves and cubic H-Bézier curves are presented.The conditions leading to inflection points and singularities (cusps and loops) are investigated and the shape of these curves is classified.The conclusions enable us to detect inflection points and singularities and get an idea of how to preserve the fair shape when designing such curves.     

Constructing a quasi-Legendre basis based on the C-Bézier basis

In the space Γn=span｛sint,cost,1,t,t2,…,tn-2｝, a C-Bézier basis is constructed by an integral approach. However, the C-Bézier basis is not orthogonal. For some applications, we construct an orthogonal basis based on the C-Bézier basis, which has remarkable properties similar to that of the Legendre basis. Then we derive the transformation matrices that convert the C-Bézier basis and the orthogonal basis into each other. As an example of application, we apply this orthogonal basis to the degree reduction approximation of the C-Bézier curves.     

Parameterization for fitting triangular mesh

In recent years, with the development of 3D data acquisition equipments, the study on reverse engineering has become more and more important. However, the existing methods for parameterization can hardly ensure that the parametric domain is rectangular, and the parametric curve grid is regular. In order to overcome these limitations, we present a novel method for parameterization of triangular meshes in this paper. The basic idea is twofold: first, because the isotherms in the steady temperature do not intersect with each other, and are distributed uniformly, no singularity (fold-over) exists in the parameterization; second, a 3D harmonic equation is solved by the finite element method to obtain the steady temperature field on a 2D triangular mesh surface with four boundaries. Therefore, our proposed method avoids the embarrassment that it is impossible to solve the 2D quasi-harmonic equation on the 2D triangular mesh without the parametric values at mesh vertices. Furthermore, the isotherms on the temperature field are taken as a set of iso-parametric curves on the triangular mesh surface. The other set of iso-parametric curves can be obtained by connecting the points with the same chord-length on the isotherms sequentially. The obtained parametric curve grid is regular, and distributed uniformly, and can map the triangular mesh surface to the unit square domain with boundaries of mesh surface to boundaries of parametric domain, which ensures that the triangular mesh surface or point cloud can be fitted with the NURBS surface.     

Shape blending of artistic brushstroke represented by disk B-spline curves

This paper presents an algorithm for automatically generating in-between frames of two artistic brushstrokes. The basic idea of the algorithm is to represent the two key frames of artistic brushstrokes in disk B-spline curves, and then make blending of their geometric intrinsic variables. Given two key frames of artistic brushstrokes, the skeleton curves can be obtained by certain skeleton-based techniques. After disk B-spline representation of the key frames is generated, interpolation of the intrinsic variables of the initial and the target disk B-spline curves is carried out. Examples show that this method can efficiently create in-between frames of artistic brushstrokes.     

Hodograph computation and bound estimation for rational B-spline curves

It is necessary to compute the derivative end estimate the bound of rational B-spline curves in design system,which has not been studied to date.To improve the function of computer aided design (CAD) system,end to enhance the efficiency of different algo- rithms of rational B-spline curves,the representation of scaled hodograph and bound of derivative magnitude of uniform planar rational B- spline curves are derived by applying Dir function,which indicates the direction of Cartesian vector between homogeneous points,discrete B-spline theory end the formula of translating the product into a summation of B-spline functions.As an application of the result above, upper bound of parametric distance between any two points in a uniform planar rational B-spline curve is further presented.     

Semi-stationary subdivision operators in geometric modeling

Based on the view of operator, a novel uniform subdivision construction method is proposed for free form curve and surface design. This method can give an appropriate manner of parameter change in the subdivision iteration with fewer parameters and better shape control, such as building local revolving surfaces. The convergent property of order 2 subdivision surface is elegantly analyzed using computing techniques of matrix. This method is promised to be valuable in Computer Aided Design and computer graphics, due to the simplicity both in mathematical theory and practical implementation, the similarity to the B-spline curve and surface, G1 continuity, the affine invariability and local flexible control.     

A two-stage algorithm of adaptive model-based parameter estimation and its application in the fast calculation of Green''s function in multi-layer soils

A two-stage algorithm is presented to determine an analytical model-based rational function by adaptively selecting sampling points. At the first stage, the rational interpolation function is created by using Stoer-Bulirsch algorithm and comparing the fitted model with the different ranks. At the second stage, the fitted model is validated by comparing with the first-principle model at a set of checking points. Some checking points will be added to the sampling series determined in the first stage to improve the fitted model. The new proposed algorithm is more accurate and effective than that using Stoer-Bulirsch interpolation only due to the fact that it appends the validation approach. When a highly oscillating curve is approximated, the validation ensures the fitting precision. With the two-stage MBPE algorithm, some time-consuming electromagnetic computations can be replaced by the interpolation from the fitting functions, which can increase the computational efficiency greatly. This method is used to calculate the samplings of the Green's functions in a multi-layer horizontal soil to demonstrate its effectiveness. The number of samples used to accurately model such Green's functions is less than 20 for the typical soil parameters. The computational time can be reduced by more than two orders of magnitude compared with that using the conventional uniform sampling technology.     

Design of perfect reconstruction rational sampling filter banks

The design of rational sampling filter banks based on a recombination structure can be formulated as a problem with two objective functions to be optimized. A new hybrid optimization method for designing perfectreconstruction rational sampling filter banks is presented, which can be used to solve a class of problems with two objective functions. This method is of good convergence and mezzo calculation cost. Satisfactory results free of aliasing in analysis and synthesis filters can be obtained by the proposed method.     

Method of So.Sensor Modeling for Fermentation Process Based on Geometric Support Vector Regression

The soft-sensor modeling for fermentation process based on standard support vector regression(SVR) needs to solve the quadratic programming problem(QPP) which will often lead to large computational burdens, slow convergence rate, low solving efficiency, and etc. In order to overcome these problems, a method of soft-sensor modeling for fermentation process based on geometric SVR is presented. In the method, the problem of solving the SVR soft-sensor model is converted into the problem of finding the nearest points between two convex hulls (CHs) or reduced convex hulls (RCHs) in geometry. Then a geometric algorithm is adopted to generate soft-sensor models of fermentation process efficiently. Furthermore, a swarm energy conservation particle swarm optimization (SEC-PSO) algorithm is proposed to seek the optimal parameters of the augmented training sample sets, the RCH size, and the kernel function which are involved in geometric SVR modeling. The method is applied to the soft-sensor modeling for a penicillin fermentation process. The experimental results show that, compared with the method based on the standard SVR, the proposed method of soft-sensor modeling based on geometric SVR for fermentation process can generate accurate soft-sensor models and has much less amount of computation, faster convergence rate, and higher efficiency.     

Extended Logistic Chaotic Sequence and Its Performance Analysis

In order to improve performance and security of image encryption algorithm effectively based on chaotic sequences, an extended chaotic sequence generating method is presented based on logistic chaotic system using Bernstein form Bézier curve generating algorithm. In order to test the pseudorandom performance of the extended chaotic sequence, we also analyze random performance, autocorrelation performance, and balance performance of the extended chaotic sequence. Simulation results show that the extended chaotic sequence generated using our method is pseudorandom and its correlation performance and balance performance are good. As an application, we apply the extended chaotic sequence in image encryption algorithm, the simulation results show that the performance of the encrypted image using our method is better than that using logistic chaotic sequence.