C˜2 Continuous Cubic Hermite Interpolation Splines with Second-Order Elliptic Variation

2021 ◽  
Vol 7 (6) ◽  
pp. 6317-6331
Author(s):  
Jie Li ◽  
Yaoyao Tu ◽  
Shilong Fei
Keyword(s):  
Shape Control ◽  
Spline Function ◽  
Hermite Interpolation ◽  
Second Order ◽  
Basis Functions ◽  
Interpolation Spline ◽  
Selection Scheme ◽  
Spline Curve ◽  
Spline Curves ◽  
Two Parameters

In order to solve the deficiency of Hermite interpolation spline with second-order elliptic variation in shape control and continuity, c-2 continuous cubic Hermite interpolation spline with second-order elliptic variation was designed. A set of cubic Hermite basis functions with two parameters was constructed. According to this set of basis functions, the three-order Hermite interpolation spline curves were defined in segments 02, and the parameter selection scheme was discussed. The corresponding cubic Hermite interpolation spline function was studied, and the method to determine the residual term and the best interpolation function was given. The results of an example show that when the interpolation conditions remain unchanged, the cubic Hermite interpolation spline curves not only reach 02 continuity, but also can use the parameters to control the shape of the curves locally or globally. By determining the best values of the parameters, the cubic Hermite interpolation spline function can get a better interpolation effect, and the smoothness of the interpolation spline curve is the best.

scholarly journals Adjustable Piecewise Quartic Hermite Spline Curve with Parameters

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Jin Xie ◽  
Xiaoyan Liu
Keyword(s):  
Shape Control ◽  
Basis Functions ◽  
Arc Length ◽  
Spline Curve ◽  
Shortest Arc ◽  
Spline Curves ◽  
Hermite Spline ◽  
C1 Continuity

In this paper, the quartic Hermite parametric interpolating spline curves are formed with the quartic Hermite basis functions with parameters, the parameter selections of the spline curves are investigated, and the criteria for the curve with the shortest arc length and the smoothest curve are given. When the interpolation conditions are set, the proposed spline curves not only achieve C1-continuity but also can realize shape control by choosing suitable parameters, which addressed the weakness of the classical cubic Hermite interpolating spline curves.

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 The EH Interpolation Spline and Its Approximation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Jin Xie ◽  
Xiaoyan Liu
Keyword(s):  
Hermite Interpolation ◽  
Single Parameter ◽  
Interpolation Spline ◽  
Two Parameters ◽  
Better Than

A new interpolation spline with two parameters, called EH interpolation spline, is presented in this paper, which is the extension of the standard cubic Hermite interpolation spline, and inherits the same properties of the standard cubic Hermite interpolation spline. Given the fixed interpolation conditions, the shape of the proposed splines can be adjusted by changing the values of the parameters. Also, the introduced spline could approximate to the interpolated function better than the standard cubic Hermite interpolation spline and the quartic Hermite interpolation splines with single parameter by a new algorithm.

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.

scholarly journals Efficient Hybrid Block Method For The Numerical Solution Of Second-order Partial Differential Problems via the Method of Lines

2021 ◽  
pp. 26-37
Author(s):  
Olumide O. Olaiya ◽  
Rasaq A. Azeez ◽  
Mark I. Modebei
Keyword(s):  
Power Series ◽  
Method Of Lines ◽  
Second Order ◽  
Basis Functions ◽  
Direct Solution ◽  
Block Method ◽  
Partial Differential ◽  
Integration Schemes ◽  
The Method Of Lines ◽  
Grid Points

This study is therefore aimed at developing classes of efficient numerical integration schemes, for direct solution of second-order Partial Differential Equations (PDEs) with the aid of the method of lines. The power series polynomials were used as basis functions for trial solutions in the derivation of the proposed schemes via collocation and interpolation techniques at some appropriately chosen grid and off-grid points the derivedschemes are consistent, zero-stable and convergent. the proposed methods perform better in terms of accuracy than some existing methods in the literature.

scholarly journals B-splines Algorithms for Solving Fredholm Linear Integro-Differential Equations

2004 ◽  
Vol 1 (2) ◽  
pp. 340-346
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equations ◽  
Spline Function ◽  
Cubic Spline ◽  
Second Order ◽  
Third Order ◽  
B Spline ◽  
Numerical Examples ◽  
B Splines ◽  
The Third ◽  
New Procedures

Algorithms using the second order of B -splines [B (x)] and the third order of B -splines [B,3(x)] are derived to solve 1' , 2nd and 3rd linear Fredholm integro-differential equations (F1DEs). These new procedures have all the useful properties of B -spline function and can be used comparatively greater computational ease and efficiency.The results of these algorithms are compared with the cubic spline function.Two numerical examples are given for conciliated the results of this method.

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