scholarly journals New Trigonometric Basis Possessing Denominator Shape Parameters

2018 ◽  
Vol 2018 ◽  
pp. 1-25 ◽  
Author(s):  
Kai Wang ◽  
Guicang Zhang
Keyword(s):  
Basis Function ◽  
Shape Parameters ◽  
B Spline ◽  
Triangular Domain ◽  
New Class ◽  
Totally Positive ◽  
Product Method ◽  
Special Value ◽  
Cutting Algorithm ◽  
Parameter Values

Four new trigonometric Bernstein-like bases with two denominator shape parameters (DTB-like basis) are constructed, based on which a kind of trigonometric Bézier-like curve with two denominator shape parameters (DTB-like curves) that are analogous to the cubic Bézier curves is proposed. The corner cutting algorithm for computing the DTB-like curves is given. Any arc of an ellipse or a parabola can be exactly represented by using the DTB-like curves. A new class of trigonometric B-spline-like basis function with two local denominator shape parameters (DT B-spline-like basis) is constructed according to the proposed DTB-like basis. The totally positive property of the DT B-spline-like basis is supported. For different shape parameter values, the associated trigonometric B-spline-like curves with two denominator shape parameters (DT B-spline-like curves) can be C2 continuous for a non-uniform knot vector. For a special value, the generated curves can be C(2n-1)  (n=1,2,3,…) continuous for a uniform knot vector. A kind of trigonometric B-spline-like surfaces with four denominator shape parameters (DT B-spline-like surface) is shown by using the tensor product method, and the associated DT B-spline-like surfaces can be C2 continuous for a nonuniform knot vector. When given a special value, the related surfaces can be C(2n-1)  (n=1,2,3,…) continuous for a uniform knot vector. A new class of trigonometric Bernstein–Bézier-like basis function with three denominator shape parameters (DT BB-like basis) over a triangular domain is also constructed. A de Casteljau-type algorithm is developed for computing the associated trigonometric Bernstein–Bézier-like patch with three denominator shape parameters (DT BB-like patch). The condition for G1 continuous jointing two DT BB-like patches over the triangular domain is deduced.

scholarly journals Quasi Cubic Trigonometric Curve and Surface

2019 ◽  
Author(s):  
Guicang Zhang ◽  
Kai Wang
Keyword(s):  
Partition Of Unity ◽  
Shape Features ◽  
Bernstein Basis ◽  
Order Continuity ◽  
B Spline ◽  
Spline Curve ◽  
Totally Positive ◽  
Chebyshev Space ◽  
Spline Basis ◽  
Cutting Algorithm

Firstly, a new set of Quasi-Cubic Trigonometric Bernstein basis with two tension shape parameters is constructed, and we prove that it is an optimal normalized totally basis in the framework of Quasi Extended Chebyshev space. And the Quasi-Cubic Trigonometric Bézier curve is generated by the basis function and the cutting algorithm of the curve are given, the shape features (cusp, inflection point, loop and convexity) of the Quasi-Cubic Trigonometric Bézier curve are analyzed by using envelope theory and topological mapping; Next we construct the non-uniform Quasi-Cubic Trigonometric B-spline basis by assuming the linear combination of the optimal normalized totally positive basis have partition of unity and continuity, and its expression is obtained. And the non-uniform B-spline basis is proved to have totally positive and high-order continuity. Finally, the non-uniform Quasi Cubic Trigonometric B-spline curve and surface are defined, the shape features of the non-uniform Quasi-Cubic Trigonometric B-spline curve are discussed, and the curve and surface are proved to be continuous.

scholarly journals A New Quasi Cubic Rational System with Two Parameters

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1070
Author(s):  
Ming-Xiu Tuo ◽  
Gui-Cang Zhang ◽  
Kai Wang
Keyword(s):  
Positive System ◽  
Local Shape ◽  
B Spline ◽  
Triangular Domain ◽  
Complex Curves ◽  
Curves And Surfaces ◽  
Spline Curves ◽  
Totally Positive ◽  
Two Parameters ◽  
New System

The purpose of this article is to develop a new system for the construction of curves and surfaces. Making the new system not only has excellent properties of the orthodox Bézier and the B-spline method but also has practical properties such as variation diminishing and local shape adjustability. First, a new set of the quasi-cubic rational (QCR) system with two parameters is given, which is verified on an optimal normalized totally positive system (B-system). The related QCR Bézier curve is defined, and the de Casteljau-type algorithm are given. Next, a group of non-uniform QCR B-spline system is shown based on the linear combination of the proposed QCR system, the relative properties of the B-spline system are analyzed. Then, the definition and properties of non-uniform QCR B-spline curves are discussed in detail. Finally, the proposed QCR system is extended to the triangular domain, which is called the quasi-cubic rational Bernstein-Bézier (QCR-BB) system, and its related definition and properties of patches are given at length. The experimental image obtained by using MATLAB shows that the newly constructed system has excellent properties such as symmetry, totally positive, and C 2 continuity, and its corresponding curve has the properties of local shape adjustability and C 2 continuity. These extended systems in the extended triangular domain have symmetry, linear independence, etc. Hence, the methods in this article are suitable for the modeling design of complex curves and surfaces.

scholarly journals Total Positivity of the Cubic Trigonometric Bézier Basis

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Xuli Han ◽  
Yuanpeng Zhu
Keyword(s):  
General Framework ◽  
Total Positivity ◽  
Point Of View ◽  
Shape Parameters ◽  
Positive Basis ◽  
Corner Cutting ◽  
Curve Design ◽  
Totally Positive ◽  
Chebyshev Space ◽  
Cutting Algorithm

Within the general framework of Quasi Extended Chebyshev space, we prove that the cubic trigonometric Bézier basis with two shape parametersλandμgiven in Han et al. (2009) forms an optimal normalized totally positive basis forλ,μ∈(-2,1]. Moreover, we show that forλ=-2orμ=-2the basis is not suited for curve design from the blossom point of view. In order to compute the corresponding cubic trigonometric Bézier curves stably and efficiently, we also develop a new corner cutting algorithm.

scholarly journals Cubic B-Spline Curves with Shape Parameter and Their Applications

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Houjun Hang ◽  
Xing Yao ◽  
Qingqing Li ◽  
Michel Artiles
Keyword(s):  
Basis Function ◽  
Shape Parameter ◽  
Engineering Application ◽  
Shape Design ◽  
Control Points ◽  
Shape Parameters ◽  
B Spline ◽  
Spline Curves ◽  
Smooth Connection

The present studies on the extension of B-spline mainly focus on Bezier methods and uniform B-spline and are confined to the adjustment role of shape parameters to curves. Researchers pay little attention to nonuniform B-spline. This paper discusses deeply the extension of the quasi-uniform B-spline curves. Firstly, by introducing shape parameters in the basis function, the spline curves are defined in matrix form. Secondly, the application of the shape parameter in shape design is analyzed deeply. With shape parameters, we get another means for adjusting the curves. Meanwhile, some examples are given. Thirdly, we discuss the smooth connection between adjacent B-spline segments in detail and present the adjusting methods. Without moving the control points position, through assigning appropriate value to the shape parameter, C1 continuity of combined spline curves can be realized easily. Results show that the methods are simple and suitable for the engineering application.

scholarly journals Quartic Rational Said-Ball-Like Basis with Tension Shape Parameters and Its Application

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Yuanpeng Zhu ◽  
Xuli Han ◽  
Shengjun Liu
Keyword(s):  
Error Estimate ◽  
Hermite Interpolation ◽  
Sufficient Conditions ◽  
Basis Functions ◽  
Shape Parameters ◽  
Interpolation Spline ◽  
Triangular Domain ◽  
Type Algorithm ◽  
Special Case ◽  
Monotonicity Preserving

Four new quartic rational Said-Ball-like basis functions, which include the cubic Said-Ball basis functions as a special case, are constructed in this paper. The new basis is applied to generate a class ofC1continuous quartic rational Hermite interpolation splines with local tension shape parameters. The error estimate expression of the proposed interpolant is given and the sufficient conditions are derived for constructing aC1positivity- or monotonicity- preserving interpolation spline. In addition, we extend the quartic rational Said-Ball-like basis to a triangular domain which has three tension shape parameters and includes the cubic triangular Said-Ball basis as a special case. In order to compute the corresponding patch stably and efficiently, a new de Casteljau-type algorithm is developed. Moreover, theG1continuous conditions are deduced for the joining of two patches.

scholarly journals Geometric Modeling Using New Cubic Trigonometric B-Spline Functions with Shape Parameter

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2102
Author(s):  
Abdul Majeed ◽  
Muhammad Abbas ◽  
Faiza Qayyum ◽  
Kenjiro T. Miura ◽  
Md Yushalify Misro ◽  
...  
Keyword(s):  
Geometric Modeling ◽  
Shape Parameter ◽  
3D Models ◽  
Basis Functions ◽  
Shape Parameters ◽  
B Spline ◽  
Spline Curves ◽  
Spline Basis ◽  
Cubic Polynomial ◽  
2D And 3D

Trigonometric B-spline curves with shape parameters are equally important and useful for modeling in Computer-Aided Geometric Design (CAGD) like classical B-spline curves. This paper introduces the cubic polynomial and rational cubic B-spline curves using new cubic basis functions with shape parameter ξ∈[0,4]. All geometric characteristics of the proposed Trigonometric B-spline curves are similar to the classical B-spline, but the shape-adjustable is additional quality that the classical B-spline curves does not hold. The properties of these bases are similar to classical B-spline basis and have been delineated. Furthermore, uniform and non-uniform rational B-spline basis are also presented. C3 and C5 continuities for trigonometric B-spline basis and C3 continuities for rational basis are derived. In order to legitimize our proposed scheme for both basis, floating and periodic curves are constructed. 2D and 3D models are also constructed using proposed curves.

scholarly journals Parameterization Method on B-Spline Curve

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
H. Haron ◽  
A. Rehman ◽  
D. I. S. Adi ◽  
S. P. Lim ◽  
T. Saba
Keyword(s):  
Computer Model ◽  
Basis Function ◽  
Real Object ◽  
Reconstruction Process ◽  
B Spline ◽  
Parameterization Method ◽  
Spline Basis ◽  
Data Points ◽  
Data Point ◽  
2D And 3D

The use of computer graphics in many areas allows a real object to be transformed into a three-dimensional computer model (3D) by developing tools to improve the visualization of two-dimensional (2D) and 3D data from series of data point. The tools involved the representation of 2D and 3D primitive entities and parameterization method using B-spline interpolation. However, there is no parameterization method which can handle all types of data points such as collinear data points and large distance of two consecutive data points. Therefore, this paper presents a new parameterization method that is able to solve those drawbacks by visualizing the 2D primitive entity of scanned data point of a real object and construct 3D computer model. The new method has improved a hybrid method by introducing exponential parameterization method in the beginning of the reconstruction process, followed by computing B-spline basis function to find maximum value of the function. The improvement includes solving a linear system of the B-spline basis function using numerical method. Improper selection of the parameterization method may lead to the singularity matrix of the system linear equations. The experimental result on different datasets show that the proposed method performs better in constructing the collinear and two consecutive data points compared to few parameterization methods.

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