scholarly journals Progressive Iterative Approximation for Extended B-Spline Interpolation Surfaces

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Yeqing Yi ◽  
Zixuan Tang ◽  
Chengzhi Liu
Keyword(s):  
Convergence Rate ◽  
Tensor Product ◽  
Spline Interpolation ◽  
Optimal Shape ◽  
Data Interpolation ◽  
Shape Parameters ◽  
Iterative Approximation ◽  
B Spline ◽  
Numerical Examples ◽  
Spline Surfaces

In order to improve the computational efficiency of data interpolation, we study the progressive iterative approximation (PIA) for tensor product extended cubic uniform B-spline surfaces. By solving the optimal shape parameters, we can minimize the spectral radius of PIA’s iteration matrix, and hence the convergence rate of PIA is accelerated. Stated numerical examples show that the optimal shape parameters make the PIA have the fastest convergence rate.

scholarly journals Analysis and Application of Quadratic B-spline Interpolation for Boundary Value Problems

2020 ◽  
pp. 11-21
Author(s):  
Md. Asaduzzaman ◽  
Liton Chandra Roy ◽  
Md. Musa Miah
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Spline Interpolation ◽  
Boundary Value ◽  
Data Interpolation ◽  
B Spline ◽  
B Splines ◽  
Numerical Example

B-splines interpolations are very popular tools for interpolating the differential equations under boundary conditions which were pioneered by Maria et.al.[16] allowing us to approximate the ordinary differential equations (ODE). The purpose of this manuscript is to analyze and test the applicability of quadratic B-spline in ODE with data interpolation, and the solving of boundary value problems. A numerical example has been given and the error in comparison with the exact value has been shown in tabulated form, and also graphical representations are shown. Maple soft and MATLAB 7.0 are used here to calculate the numerical results and to represent the comparative graphs.

scholarly journals Innovative tooth contact analysis with non-uniform rational b-spline surfaces

2021 ◽  
Author(s):  
Felix Müller ◽  
Stefan Schumann ◽  
Berthold Schlecht
Keyword(s):  
Tensor Product ◽  
Measurement Data ◽  
Software Tool ◽  
Contact Analysis ◽  
Tooth Root ◽  
Surface Representation ◽  
Tooth Contact Analysis ◽  
Tooth Contact ◽  
B Spline ◽  
Spline Surfaces

AbstractMore and more simulation tools are being used in the development of gears in order to save development time and costs while improving the gears. BECAL is a comprehensive software tool for the tooth contact analysis (TCA) of bevel, hypoid, beveloid and spur gears. The gear geometry is provided by a manufacturing simulation or a geometry import. To determine the exact contact conditions in the TCA, the discrete flank points are converted into a continuous and differentiable surface representation. At present, it is an approximation by means of Bézier tensor product surfaces. With this surface representation, significant deviations to the target points can occur depending on the tooth geometry. In particular tip, root and end relief, strongly curved tooth root geometries or discontinuous topological measurement data due to e.g. micro-pitting can only be considered insufficiently.Hence, a new method for surface approximation with non-uniform rational b‑spline surfaces (NURBS) is presented. Its application can significantly improve the surface representation compared to the target geometry, leading to more realistic results regarding contact stress, tooth root stress and transmission error. To illustrate the advantages, NURBS-based surfaces are compared with the Bézier tensor product surfaces. Finally, the potential of the new approach regarding the prediction of lifetime and acoustics is demonstrated by application to different gear geometries.

Knot line refinement algorithms for tensor product B-spline surfaces

1985 ◽  
Vol 2 (1-3) ◽  
pp. 133-139 ◽  
Author(s):  
T. Lyche ◽  
E. Cohen ◽  
K. Mørken
Keyword(s):  
Tensor Product ◽  
B Spline ◽  
Spline Surfaces

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.

PIECEWISE QUARTIC TRIGONOMETRIC POLYNOMIAL B-SPLINE CURVES WITH TWO SHAPE PARAMETERS

2012 ◽  
Vol 12 (04) ◽  
pp. 1250028
Author(s):  
MRIDULA DUBE ◽  
REENU SHARMA
Keyword(s):  
Trigonometric Polynomial ◽  
Shape Parameters ◽  
B Spline ◽  
Spline Curve ◽  
Control Polygon ◽  
Spline Curves ◽  
Spline Surfaces ◽  
Polynomial Curve ◽  
Trigonometric Spline ◽  
The Given

Analogous to the quartic B-splines curve, a piecewise quartic trigonometric polynomial B-spline curve with two shape parameters is presented in this paper. Each curve segment is generated by three consecutive control points. The given curve posses many properties of the B-spline curve. These curves are closer to the control polygon than the different other curves considered in this paper, for different values of shape parameters for each curve. With the increase of the value of shape parameters, the curve approach to the control polygon. For nonuniform and uniform knot vector the given curves have C0, G3; C1, G3; C1, G7; and C3 continuity for different choice of shape parameters. A quartic trigonometric Bézier curves are also introduced as a special case of the given trigonometric spline curves. A comparison of quartic trigonometric polynomial curve is made with different other curves. In the last, quartic trigonometric spline surfaces with two shape parameters are constructed. They have most properties of the corresponding curves.

Geometric Design and Fabrication of Developable Bezier and B-spline Surfaces

1994 ◽  
Vol 116 (4) ◽  
pp. 1042-1048 ◽  
Author(s):  
R. M. C. Bodduluri ◽  
B. Ravani
Keyword(s):  
Spline Interpolation ◽  
Geometric Design ◽  
Developable Surface ◽  
Knot Insertion ◽  
B Spline ◽  
Developable Surfaces ◽  
Spline Surfaces ◽  
Insertion Algorithm ◽  
Cad Cam ◽  
Type Construction

In this paper we study Computer Aided Geometric Design (CAGD) and Manufacturing (CAM) of developable surfaces. We develop a direct representation of developable surfaces in terms of plane geometry. It uses control planes to determine a surface which is a Bezier or a B-spline interpolation of the control planes. In the Bezier case, a de Casteljau type construction method is presented for geometric design of developable Bezier surfaces. In the B-spline case, de Boor type construction for the geometric design of the developable surface and Boehm type knot insertion algorithm are presented. In the area of manufacturing or fabrication of developable surfaces, we present simple methods for both development of a surface into a plane and bending of a flat plane into a desired developable surface. The approach presented uses plane and line geometries and eliminates the need for solving differential equations of Riccatti type used in previous methods. The results are illustrated using an example generated by a CAD/CAM system implemented based on the theory presented.

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