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.

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.

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 Interpolating Industrial Robot Orientation with Hermite Spline Curve Based on Logarithmic Quaternion

2019 ◽  
Vol 37 (6) ◽  
pp. 1165-1173
Author(s):  
Yasong Pu ◽  
Yaoyao Shi ◽  
Xiaojun Lin ◽  
Jian Guo
Keyword(s):  
Industrial Robot ◽  
B Spline ◽  
Spline Curve ◽  
Cartesian Space ◽  
Spline Curves ◽  
Detailed Method ◽  
Quaternion Space ◽  
Hermite Spline ◽  
Bézier Spline ◽  
Working Quality

Smooth orientation planning has an important influence on the working quality and service life as for industrial robot. Based on the logarithmic quaternion, a compact method to map a spline curve from Cartesian space to quaternion space is proposed, and consequently the multi-orientation smooth interpolation of quaternion is realized. Combining with the relevant example case, the detailed method and steps of multi-orientation interpolation are introduced for mapping Hermite spline curve into quaternion space, and the validity of the principle is verified by using the example case. The present multi-orientation smooth interpolation of quaternion has the characteristics of simple construction, easy implementation and intuitive understanding. The method is not only applicable to multi-orientation interpolation of quaternion with Hermite spline curve, but also can extended to the spline curves such as Bezier spline and B-spline.

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.

T-B Spline Curves with a Shape Parameter

2012 ◽  
Vol 241-244 ◽  
pp. 2144-2148
Author(s):  
Li Juan Chen ◽  
Ming Zhu Li
Keyword(s):  
Shape Parameter ◽  
Simple Structure ◽  
Control Points ◽  
Curve Segment ◽  
Closed Curves ◽  
B Spline ◽  
Spline Curve ◽  
Spline Curves

A T-B spline curves with a shape parameter λ is presented in this paper, which has simple structure and can be used to design curves. Analogous to the four B-spline curves, each curve segment is generated by five consecutive control points. For equidistant knots, the curves are C^2 continuous, but when the shape parameter λ equals to 0 , the curves are C^3 continuous. Moreover, this spline curve can be used to construct open and closed curves and can express ellipses conveniently.

scholarly journals Optimal Approximation of Biquartic Polynomials by Bicubic Splines

2018 ◽  
Vol 173 ◽  
pp. 03012
Author(s):  
Viliam Kačala ◽  
Csaba Török
Keyword(s):  
Optimal Approximation ◽  
First Derivative ◽  
Polynomial Coefficients ◽  
Spline Curves ◽  
Spline Surfaces ◽  
Chebyshev Norm ◽  
Quartic Polynomial ◽  
Hermite Spline ◽  
Open Question ◽  
Bicubic Splines

Recently an unexpected approximation property between polynomials of degree three and four was revealed within the framework of two-part approximation models in 2-norm, Chebyshev norm and Holladay seminorm. Namely, it was proved that if a two-component cubic Hermite spline’s first derivative at the shared knot is computed from the first derivative of a quartic polynomial, then the spline is a clamped spline of classC2and also the best approximant to the polynomial.Although it was known that a 2 × 2 component uniform bicubic Hermite spline is a clamped spline of classC2if the derivatives at the shared knots are given by the first derivatives of a biquartic polynomial, the optimality of such approximation remained an open question.The goal of this paper is to resolve this problem. Unlike the spline curves, in the case of spline surfaces it is insufficient to suppose that the grid should be uniform and the spline derivatives computed from a biquartic polynomial. We show that the biquartic polynomial coefficients have to satisfy some additional constraints to achieve optimal approximation by bicubic splines.

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.

Approximate Merging B-Spline Curves via Least Square Approximation

2014 ◽  
Vol 556-562 ◽  
pp. 3496-3500 ◽  
Author(s):  
Si Hui Shu ◽  
Zi Zhi Lin
Keyword(s):  
Approximation Error ◽  
Least Square ◽  
Error Tolerance ◽  
Knot Insertion ◽  
Least Squares Approximation ◽  
B Spline ◽  
Spline Curve ◽  
Spline Curves ◽  
Insertion Algorithm ◽  
Approximate Merging

An algorithm of B-spline curve approximate merging of two adjacent B-spline curves is presented in this paper. In this algorithm, the approximation error between two curves is computed using norm which is known as best least square approximation. We develop a method based on weighed and constrained least squares approximation, which adds a weight function in object function to reduce error of merging. The knot insertion algorithm is also developed to meet the error tolerance.

Analysis on Quality Evaluation Criteria for Spline Curves

2013 ◽  
Vol 365-366 ◽  
pp. 99-103
Author(s):  
Shi Jun Ji ◽  
Ji Zhao ◽  
Lei Zhang ◽  
Hui Juan Yu ◽  
Xiao Long Liu ◽  
...  
Keyword(s):  
Curve Fitting ◽  
Geometric Modeling ◽  
Quality Evaluation ◽  
Evaluation Criteria ◽  
Affecting Factors ◽  
Spline Curve ◽  
Spline Curves ◽  
Fitting Method ◽  
Cad Cam ◽  
Curve Fitting Method

With the development of CAD/CAM, reverse engineering, geometric modeling, animation and computer graphics. In the past fifty years, many kinds of spline curve fitting methods have been produced. But, to authors knowledge, quality evaluation criterion about different spline curves are still not established and there is a lack of a comparing standard for analyzing different spline curves. Two spline curve fitting method comparing theorem is given and several various spline curves affecting factors are analyzed in this paper. Some comparable results are obtained by using the proposed quality evaluating theorem to different spline curves.

G1/C1 Matching of Spline Curves

2010 ◽  
Vol 20-23 ◽  
pp. 202-208
Author(s):  
Xu Min Liu ◽  
Wei Xiang Xu ◽  
Jing Xu ◽  
Yong Guan
Keyword(s):  
Matching Condition ◽  
Basic Function ◽  
B Spline ◽  
Spline Curve ◽  
Spline Curves

The research is mainly made on the G1/C1 matching condition of spline curves. On the basis of the analysis on the basic function of T-B spline curves and the features of curve endpoints, we proposed the n+1 order T-B spline basic function and the solving method. The G1/C1 matching condition of C-B spline curves and T-B spline curves is put forward in this paper. On this condition, when matching C-B spline curves and T-B spline curves, the controlling vertexes can be added to make C-B spline curve tangent with the first and last edge by the first and last vertex of controlling polygon. Application instances were put up in this paper which illustrated that the G1/C1 matching between T-B spline curve and C-B spline curve using the feature of T-B spline curve which can represents semiellipse arc (semicircle arc) precisely can solve the problem that C-B spline curve cannot represents semiellipse arc (semicircle arc) precisely.

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