scholarly journals Cubic Trigonometric Nonuniform Spline Curves and Surfaces

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Lanlan Yan
Keyword(s):  
Shape Parameter ◽  
Basis Functions ◽  
Surfaces Of Revolution ◽  
Local Shape ◽  
Spline Curves ◽  
Spline Surfaces ◽  
Totally Positive ◽  
Positive Property ◽  
Spline Basis ◽  
Representation Scheme

A class of cubic trigonometric nonuniform spline basis functions with a local shape parameter is constructed. Their totally positive property is proved. The associated spline curves inherit most properties of usual polynomialB-spline curves and enjoy some other advantageous properties for engineering design. They haveC2continuity at single knots. For equidistant knots, they haveC3continuity andC5continuity for particular choice of shape parameter. They can express freeform curves as well as ellipses. The associated spline surfaces can exactly represent the surfaces of revolution. Thus the curve and surface representation scheme unifies the representation of freeform shape and some analytical shapes, which is popular in engineering.

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 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.

Cubic Uniform B-Spline Curve and Surface with Multiple Shape Parameters

2014 ◽  
Vol 543-547 ◽  
pp. 1860-1863
Author(s):  
Xi Wang ◽  
Cui Cui Gao ◽  
Chen Jiang
Keyword(s):  
Shape Control ◽  
Basis Functions ◽  
Control Parameters ◽  
Shape Parameters ◽  
Surface Design ◽  
Local Shape ◽  
B Spline ◽  
Spline Curves ◽  
Curve Fit ◽  
Polynomial Curve

In order to construct B-spline curves with local shape control parameters, a class of polynomial basis functions with two local shape parameters is presented. Properties of the proposed basis functions are analyzed and the corresponding piecewise polynomial curve is constructed with two local shape control parameters accordingly. In particular, the G1 continuous and the shapes of other segments of the curve can remain unchangeably during the manipulation on the shape of each segment on the curve. Numerical examples illustrate that the constructed curve fit to the control polygon very well. Furthermore, its applications in curve design is discussed and an extend application on surface design is also presented. Modeling examples show that the new curve is very valuable for the design of curves and surfaces.

Generation and Modification of Non-Uniform B-Spline Surface Approximations to PDE Surfaces Using the Finite Element Method

1990 ◽  
Author(s):  
Joanna M. Brown ◽  
Malcolm I. G. Bloor ◽  
M. Susan Bloor ◽  
Michael J. Wilson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Spline Approximation ◽  
The Finite Element Method ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Spline Basis ◽  
Element Method

Abstract A PDE surface is generated by solving partial differential equations subject to boundary conditions. To obtain an approximation of the PDE surface in the form of a B-spline surface the finite element method, with the basis formed from B-spline basis functions, can be used to solve the equations. The procedure is simplest when uniform B-splines are used, but it is also feasible, and in some cases desirable, to use non-uniform B-splines. It will also be shown that it is possible, if required, to modify the non-uniform B-spline approximation in a variety of ways, using the properties of B-spline surfaces.

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.

Isogeometric analysis of laminated smart shell structures covered with piezoelectric sensors and actuators using degenerated shell formulation

2019 ◽  
Vol 30 (13) ◽  
pp. 1913-1931 ◽  
Author(s):  
Sajjad Nikoei ◽  
Behrooz Hassani
Keyword(s):  
Single Layer ◽  
Laminated Composite ◽  
Piezoelectric Sensor ◽  
Basis Functions ◽  
Shell Structures ◽  
Free Form ◽  
Control Analysis ◽  
Sensors And Actuators ◽  
Spline Basis ◽  
Shell Formulation

An isogeometric approach to the analysis of laminated composite smart shell structures based on the degenerated formulation and Mindlin–Reissner assumptions using non-uniform rational B-spline basis functions is the subject of this article. To model the laminated orthotropic smart free-form shells, the equivalent single layer theory is adopted, and an accurate approach to construct the local basis systems is used. To consider the electric potential in the piezoelectric layers, a sub-layer approach is employed that assumes linear variation over the thickness of the sub-layer. To investigate the performance of the approach, static, free vibration, and static control analysis of laminated composite shells covered with piezoelectric sensor and actuator layers with different degrees of basis functions is performed. Also, the effect of mechanical loading, various input voltages, and different boundary conditions on the static response and natural frequencies have been investigated. Several numerical examples are presented to demonstrate the efficiency and accuracy of the approach and validated with the existing results from the literature.

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