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Geometric construction of quintic parametric B-splines
Abstract
The aim of this paper is to present a new class of B-spline-like functions with tension properties. The main feature of these basis functions consists in possessing C3 or even C4 continuity and, at the same time, being endowed by shape parameters that can be easily handled. Therefore they constitute a useful tool for the construction of curves satisfying some prescribed shape constraints. The construction is based on a geometric approach which uses parametric curves with piecewise quintic components.
... The approach enables a circle arc to be approximated accurately but does not deal with the problem of the sequence of straight lines and circle arcs. Other works propose curve construction methodologies using tension properties [22]. However, these methods are not directly usable in the case of pocket machining, because the path presents too many discontinuities. ...
This article proposes a new method of 2D curve interpolation using non-uniform cubic B-splines particularly adapted to the interpolation of sequences of straight lines and circle arcs. The purpose of this method is to calculate C2 continuous curves adapted to high feedrate pocket machining. Industrially machined pockets usually present simple forms. Generally, the tool path is defined by circle arcs and line segments that introduce slowdowns during machining. Thus, a method for approximating a sequence of line segments and circle arcs using Bspline curves is proposed. The proposed method ensures exact line interpolation, to approach the tool path precisely, to reduce the number of control points and to avoid thickening and oscillation at the connections between line segments and circle arcs. Various applications are presented and numerous tests on machine tools allow the advantages of this method to be illustrated.
A neural network scheme based on quintic B-spline functions of optimality systems arising from linear parabolic optimal control problems is discussed. It is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to the optimal solution of the original problem. Theoretical and experimental results show the advantages of B-spline functions and demonstrate that the suggested scheme is able to efficiently solve constrained optimal control problems. The resulting scheme also shows that robustness with respect to changes of the value of , the weight of the cost of the control, is sufficiently small.
A survey is given of algorithms for passing a curve through data points so as to preserve the shape of the data.
This book is based on the author's experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shape-preserving properties of the B-spline series.
We present a geometric construction of a two-parameter family of C 3 , 4-interval supported, B-spline like functions obtained by the parametric approach, using planar curves with piecewise quintic components and geometric continuity of order 3. The parameters act as shape parameters and have a clear geometric interpretation.
We present a B-spline-Bézier representation of geometrically continuous quartic and quintic spline curves based on the Bézier representation of these curves described in a foregoing paper of the two authors. The influence of the design parameters is illustrated, and local support basis functions are given. Geometrically continuous curves include as special cases visually continuous curves as well as minimizing spline curves due to Nielson [1] and Hagen [2].
In this paper a class of C2∩FC3 spline curves possessing tension properties is described. These curves can be constructed using a simple modification of the well-known geometric construction of C4 quintic splines; therefore their shape can be easily controlled using the control net. Their applications in approximation and interpolation of spatial data will be discussed.
The paper proposes a method for the construction of C
2 quasi-interpolating functions with tension properties. The constructed quasi-interpolant is a parametric cubic curve and its shape can be easily controlled via tension parameters which have an immediate geometric interpretation. Numerical examples are presented.
A classic reference and text, this book introduces the foundations used to create an accurate computer screen image using mathematical tools. This comprehensive guide is a handbook for students and practitioners and includes an extensive bibliography for further study.
We present a new method for the construction of shape-preserving curves approximating a given set of 3D data, based on the space of “quintic like” polynomial splines with variable degrees recently introduced in [7]. These splines – which are C3 and therefore curvature and torsion continuous – possess a very simple geometric structure, which permits to easily handle the shape-constraints.
We present C
2 quasi-interpolating schemes with tension properties. The B-splines like functions used in the quasi-interpolanting schemes are parametric cubic curves and their shape can be easily controlled via tension parameters which have an immediate geometric interpretation. Applications to the problem of approximation of curves with shape-constraints are discussed.