scholarly journals A New 5-Point Ternary Interpolating Subdivision Scheme and Its Differentiability

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Ghulam Mustafa ◽  
Jayyada Irum ◽  
Mehwish Bari
Keyword(s):  
Tensor Product ◽  
Shape Parameter ◽  
Subdivision Scheme ◽  
Smooth Curves ◽  
Numerical Examples ◽  
Interpolating Scheme

A new 5-point ternary interpolating scheme with a shape parameter is introduced. The resulting curve is for a certain range of parameters. The differentiable properties of the proposed scheme to extend its application in the generation of smooth curves are explored. Application of the proposed scheme is given to show its visual smoothness. The scheme is also extended to a 5-point tensor product ternary interpolating scheme, and its numerical examples are also included.

scholarly journals Local Convexity-PreservingC2Rational Cubic Spline for Convex Data

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Muhammad Abbas ◽  
Ahmad Abd Majid ◽  
Jamaludin Md. Ali
Keyword(s):  
Shape Parameter ◽  
Cubic Spline ◽  
Convex Curve ◽  
Shape Feature ◽  
Local Convexity ◽  
Shape Parameters ◽  
Numerical Examples ◽  
2D Data ◽  
Industrial Demand ◽  
Made In

We present the smooth and visually pleasant display of 2D data when it is convex, which is contribution towards the improvements over existing methods. This improvement can be used to get the more accurate results. An attempt has been made in order to develop the local convexity-preserving interpolant for convex data usingC2rational cubic spline. It involves three families of shape parameters in its representation. Data dependent sufficient constraints are imposed on single shape parameter to conserve the inherited shape feature of data. Remaining two of these shape parameters are used for the modification of convex curve to get a visually pleasing curve according to industrial demand. The scheme is tested through several numerical examples, showing that the scheme is local, computationally economical, and visually pleasing.

Numerical solution of two-dimensional first kind Fredholm integral equations by using linear Legendre wavelet

2016 ◽  
Vol 14 (01) ◽  
pp. 1650004 ◽  
Author(s):  
M. Tahami ◽  
A. Askari Hemmat ◽  
S. A. Yousefi
Keyword(s):  
Integral Equation ◽  
Tensor Product ◽  
Numerical Solution ◽  
Fredholm Integral Equation ◽  
Fredholm Integral Equations ◽  
Wavelet Basis ◽  
Two Dimensional ◽  
Legendre Wavelets ◽  
Numerical Examples ◽  
One Dimensional

In one-dimensional problems, the Legendre wavelets are good candidates for approximation. In this paper, we present a numerical method for solving two-dimensional first kind Fredholm integral equation. The method is based upon two-dimensional linear Legendre wavelet basis approximation. By applying tensor product of one-dimensional linear Legendre wavelet we construct a two-dimensional wavelet. Finally, we give some numerical examples.

scholarly journals A New 4-PointC3Quaternary Approximating Subdivision Scheme

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Ghulam Mustafa ◽  
Faheem Khan
Keyword(s):  
Shape Parameter ◽  
Approximation Order ◽  
Subdivision Scheme ◽  
Subdivision Schemes ◽  
Approximating Subdivision Scheme

A new 4-pointC3quaternary approximating subdivision scheme with one shape parameter is proposed and analyzed. Its smoothness and approximation order are higher but support is smaller in comparison with the existing binary and ternary 4-point subdivision schemes.

scholarly journals The Quaternary Interpolating Scheme for Geometric Design

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Shahid S. Siddiqi ◽  
Muhammad Younis
Keyword(s):  
Approximation Order ◽  
Geometric Design ◽  
Subdivision Scheme ◽  
Conic Sections ◽  
Linear Spaces ◽  
The Family ◽  
Support Size ◽  
Interpolating Scheme ◽  
Better Than

A novel 4-point interpolating subdivision scheme is presented, which generates the family of C2 limiting curves and its limiting function has support on [−7/3,7/3]. It behaves better than classical 4-point binary and ternary schemes with the same approximation order in many aspects that it has smaller support size, higher smoothness, and is computationally more efficient. The proposed nonstationary scheme can reproduce the functions of linear spaces spanned by {1,sin(αx),cos(αx)} for 0<α<π/3. Moreover, some examples are illustrated to show that the proposed scheme can also reproduce asteroids, cardioids, and conic sections as well.

scholarly journals Construction and Application of Nine-Tic B-Spline Tensor Product SS

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 675 ◽  
Author(s):  
Abdul Ghaffar ◽  
Mudassar Iqbal ◽  
Mehwish Bari ◽  
Sardar Muhammad Hussain ◽  
Raheela Manzoor ◽  
...  
Keyword(s):  
Tensor Product ◽  
Spectral Radius ◽  
Execution Time ◽  
Computer Programming ◽  
Surface Modeling ◽  
Hölder Regularity ◽  
Subdivision Scheme ◽  
Joint Spectral Radius ◽  
B Spline ◽  
Holder Regularity

In this paper, we propose and analyze a tensor product of nine-tic B-spline subdivision scheme (SS) to reduce the execution time needed to compute the subdivision process of quad meshes. We discuss some essential features of the proposed SS such as continuity, polynomial generation, joint spectral radius, holder regularity and limit stencil. Some results of the SS using surface modeling with the help of computer programming are shown.

scholarly journals Shape-Preserving Properties of a Relaxed Four-Point Interpolating Subdivision Scheme

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 806 ◽  
Author(s):  
Pakeeza Ashraf ◽  
Abdul Ghaffar ◽  
Dumitru Baleanu ◽  
Irem Sehar ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Graphical Representation ◽  
Subdivision Scheme ◽  
Control Points ◽  
Positivity Preserving ◽  
Shape Preserving ◽  
Numerical Examples ◽  
Initial Control ◽  
Point Scheme ◽  
Shape Preserving Properties ◽  
Monotonicity Preserving

In this paper, we analyze shape-preserving behavior of a relaxed four-point binary interpolating subdivision scheme. These shape-preserving properties include positivity-preserving, monotonicity-preserving and convexity-preserving. We establish the conditions on the initial control points that allow the generation of shape-preserving limit curves by the four-point scheme. Some numerical examples are given to illustrate the graphical representation of shape-preserving properties of the relaxed scheme.

A Nonlinear Ternary Circle-Preserving Interpolatory Subdivision Scheme

2013 ◽  
Vol 427-429 ◽  
pp. 2170-2173 ◽  
Author(s):  
Qian Song ◽  
Hong Chan Zheng ◽  
Guo Hua Peng
Keyword(s):  
Subdivision Scheme ◽  
Limit Curve ◽  
Numerical Examples ◽  
Interpolatory Subdivision

In this paper, we present a new nonlinear ternary interpolatory subdivision scheme which has the properties of convexity-preserving, circle-preserving and the limit curve iscontinuous. In each subdivision step, the newly generating points will be on the circle determined by the interpolatory point and its adjacent points. Numerical examples show that this algorithm is simple and curves generated by this subdivision scheme are fair curves.

scholarly journals Generalized 5-Point Approximating Subdivision Scheme of Varying Arity

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 474 ◽  
Author(s):  
Sardar Muhammad Hussain ◽  
Aziz Ur Rehman ◽  
Dumitru Baleanu ◽  
Kottakkaran Sooppy Nisar ◽  
Abdul Ghaffar ◽  
...  
Keyword(s):  
Error Bound ◽  
Shape Parameter ◽  
Geometric Design ◽  
Hölder Regularity ◽  
Subdivision Scheme ◽  
Subdivision Schemes ◽  
Computer Aided Geometric Design ◽  
Approximating Subdivision Scheme ◽  
Computer Aided ◽  
Holder Regularity

The Subdivision Schemes (SSs) have been the heart of Computer Aided Geometric Design (CAGD) almost from its origin, and various analyses of SSs have been conducted. SSs are commonly used in CAGD and several methods have been invented to design curves/surfaces produced by SSs to applied geometry. In this article, we consider an algorithm that generates the 5-point approximating subdivision scheme with varying arity. By applying the algorithm, we further discuss several properties: continuity, Hölder regularity, limit stencils, error bound, and shape of limit curves. The efficiency of the scheme is also depicted with assuming different values of shape parameter along with its application.

Cubic TP B-Spline Curves with a Shape Parameter

2013 ◽  
Vol 11 ◽  
pp. 59-72 ◽  
Author(s):  
Mridula Dube ◽  
Reenu Sharma
Keyword(s):  
Tensor Product ◽  
Shape Parameter ◽  
Control Points ◽  
Shape Parameters ◽  
Special Shape ◽  
B Spline ◽  
Control Polygon ◽  
Spline Curves ◽  
Trigonometric Spline ◽  
The Given

In this paper a new kind of splines, called cubic trigonometric polynomial B-spline (cubic TP B-spline) curves with a shape parameter, are constructed over the space spanned by As each piece of the curve is generated by three consecutive control points, they posses many properties of the quadratic B-spline curves. These trigonometric curves with a non-uniform knot vector are C1 and G2 continuous. They are C2 continuous when choosing special shape parameter for non-uniform knot vector. These curves are closer to the control polygon than the quadratic B-spline curves when choosing special shape parameters. With the increase of the shape parameter, the trigonometric spline curves approximate to the control polygon. The given curves posses many properties of the quadratic B-spline curves. The generation of tensor product surfaces by these new splines is straightforward.

scholarly journals Two Extensions of the Quadratic Nonuniform B-Spline Curve with Local Shape Parameter Series

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Xiang Kong ◽  
Jun Chen
Keyword(s):  
Line Segment ◽  
Shape Parameter ◽  
Linear Functions ◽  
Local Shape ◽  
B Spline ◽  
Numerical Examples ◽  
Spline Curve ◽  
Control Polygon ◽  
Tangent Direction ◽  
Special Case

Two extensions of the quadratic nonuniform B-spline curve with local shape parameter series, called the W3D3C1P2 spline curve and the W3D4C2P1 spline curve, are introduced in the paper. The new extensions not only inherit most excellent properties of the quadratic nonuniform B-spline curve but also can move locally toward or against the fixed control polygon by varying the shape parameter series. They are C1 and C2 continuous separately. Furthermore, the W3D3C1P2 spline curve includes the quadratic nonuniform B-spline curve as a special case. Two applications, the interpolation of the position and the corresponding tangent direction and the interpolation of a line segment, are discussed without solving a system of linear functions. Several numerical examples indicated that the new extensions are valid and can easily be applied.

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