Weierstrass-type approximation theorems with Pythagorean hodograph curves

We prove the Weierstrass-type approximation theorem that states every C¹ curve in the 2-dimensional or 3-dimensional Euclidean space or in the 3-dimensional Minkowski space can be uniformly approximated by Pythagorean hodograph curves in the corresponding space. This abundance of PH curves is another theoretical confirmation of the usefulness and the versatility of the PH curves. We also address some algorithmic aspects of proposed PH approximation schemes and their convergence rates.     

Approximating curves and their offsets using biarcs and Pythagorean hodograph quintics

This paper compares two techniques for the approximation of the offsets to a given planar curve. The two methods are based on approximate conversion of the planar curve into circular splines and Pythagorean hodograph (PH) splines, respectively. The circular splines are obtained using a novel variant of biarc interpolation, while the PH splines are constructed via Hermite interpolation of C¹ boundary data.We analyze the approximation order of both conversion procedures. As a new result, the C¹ Hermite interpolation with PH quintics is shown to have approximation order 4 with respect to the original curve, and 3 with respect to its offsets. In addition, we study the resulting data volume, both for the original curve and for its offsets. It is shown that PH splines outperform the circular splines for increasing accuracy, due to the higher approximation order.     

Shape preserving least-squares approximation by polynomial parametric spline curves

This article presents a method for shape preserving least-squares approximation. This method generalizes an algorithm of Dierckx (1980, 1993) to the case of planar parametric curves. Using a reference curve we generate linear sufficient conditions for the convexity of the approximant. This leads us to a quadratic programming problem which can be solved exactly, e.g., with an active set strategy.     

Interchanging spline curves using IGES

Current CAD/CAM systems employ a variety of spline types, spline representations, and curve design algorithms. The IGES (initial graphic exchange specification) was designed to enable interchange of the more commonly used spline types among CAD/CAM systems, independent of the design algorithms originally used to create them. IGES supports parametric cubic splines through a piecewise polynomial representation. This paper describes the design considerations leading to the interchange common spline curve types. A short introduction to splines and spline representations is included.     

Impedance measurement using multiharmonic least-squares waveform fitting algorithm

In this paper, a multiharmonic least-squares waveform fitting algorithm is used to measure impedances. By using analog to digital converters, a function generator, a reference impedance and the fitting algorithm, we show that it is possible to obtain the frequency response of a unknown linear impedance at the frequencies that correspond to the harmonics of the stimulus input signal.

Results show that, with a 400 Hz triangular signal, harmonics up to the 49th can be used to measure the impedance frequency response of a parallel RLC circuit with experimental standard deviations below 1.5% for the impedance magnitude and 1° for the impedance phase.     

Cone spline approximation via fat conic spline fitting

Fat conic section and fat conic spline are defined. With well established properties of fat conic splines, the problem of approximating a ruled surface by a tangent smooth cone spline can then be changed as the problem of fitting a plane fat curve by a fat conic spline. Moreover, the fitting error between the ruled surface and the cone spline can be estimated explicitly via fat conic spline fitting. An efficient fitting algorithm is also proposed for fat conic spline fitting with controllable tolerances. Several examples about approximation of general developable surfaces or other types of ruled surfaces by cone spline surfaces are presented.     

用矩阵运算实现曲线拟合中的最小二乘法

介绍了一种用矩阵运算实现曲线拟合过程中的最小二乘法的方法 ,避免复杂的求偏导过程 ,使得曲线拟合计算变得十分容易 ,在仪器仪表和传感器标定中有很好的应用价值。文章最后并有举例说明。     

Data fitting with a spline using a real-coded genetic algorithm

To obtain a good approximation for data fitting with a spline, frequently we have to deal with knots as variables. The problem to be solved then becomes a continuous nonlinear and multivariate optimization problem with many local optima. Therefore, it is difficult to obtain the global optimum. In this paper, we propose a method for solving this problem by using a real-coded genetic algorithm. Our method can treat not only data with a smooth underlying function, but also data with an underlying function having discontinuous points and/or cusps. We search for the best model among candidate models by using the Bayes Information Criterion (BIC). With this, we can appropriately determine the number and locations of knots automatically and simultaneously. Five examples of data fitting are given to show the performance of our method.     

RETRACTED: Characterization of Pythagorean curves and Pythagoreanization using a rational transform

This article has been retracted at the request of the Editor-in-Chief and the authors.Reason: it has been found belatedly that the main result of the paper is not original to their research (see the authors' statement below). We, the editors, apologize to the readers for any confusion and we will make every effort to prevent such a regrettable case in future. In this effort, we also sincerely remind authors of their scientific duty to carry out a complete literature search before making originality claims in submitted articles.Authors' Statement: “We, the authors of the article, ‘Characterization of Pythagorean curves and Pythagoreanization using a rational transform’, Journal of Symbolic Computation 37 (2004), 377–389, were alerted by Professor Rida Farouki that the result presented in the article may have been previously obtained by Dr. Wei Lü in the article, ‘Offset-rational parametric plane curves’, Computer Aided Geometric Design 12 (1995), 601–616.We have checked it. Even though there are some differences, we agree that Wei Lü indeed has a precedence on the main result and thus he should be given the full credit on the result. Accordingly, we also agree with the editor's decision to retract the article.We regret that we have not carried out a complete literature search before embarking on our work. We apologize to Dr. Wei Lü and colleagues for any confusion caused by our mistake. We have learned the importance of a complete literature search.Most of all, we thank Professor Rida Farouki for kindly alerting us to the problem.”Min-Ho Ahn and Gwang-Il Kim     

Logistic curve fitting and parameter estimation using nonlinear noniterative least-squares regression analysis

A microcomputer program has been developed for the fitting of the logistic curve to biological, medical, and other experimental data. In addition to supplying estimates for all of the logistic curve parameters, the program provides the fitted result for each input datum thus allowing for the immediate assessment of the logistic curve and detection of possible outliers.     

Surfaces with Pythagorean normals along rational curves

A rational curve on a rational surface such that the unit normal vector field of the surface along this curve is rational will be called a curve providing Pythagorean surface normals (or shortly a PSN curve). These curves represent rational paths on the surface along which the surface possesses rational offset curves. Our aim is to study rational surfaces containing enough PSN curves. The relation with PN surfaces will be also investigated and thoroughly discussed. The algebraic and geometric properties of PSN curves will be described using the theory of double planes. The main motivation for this contribution is to bring the theory of rational offsets of rational surfaces closer to the practical problems appearing in numerical-control machining where the milling cutter does not follow continuously the whole offset surface but only certain chosen trajectories on it. A special attention will be devoted to rational surfaces with pencils of PSN curves.     

On control polygons of quartic Pythagorean–hodograph curves

In this paper, we study a necessary and sufficient condition for a planar quartic Bézier curve to possess a Pythagorean–hodograph (PH). Based on the definition of PH curve and complex representation of planar curve, we deduce geometric conditions in terms of the legs of the control polygon which guarantee the PH property. We also discuss the problem of G¹ Hermite interpolation by planar PH quartics.     

三次插值样条曲线拟合多核并行算法

充分利用多核技术提升多核处理器的资源利用率,缩短执行时间,发挥多核系统的优异性能。在多核计算机上设计了解三对角方程组的奇偶约化多线程并行程序,实现了三次样条曲线拟合的快速计算。通过实验结果的加速比对比,可以看出并行后缩短了求解方程组的时间,多核资源得到充分利用。结果表明,奇偶约化多核并行算法在三次样条曲线拟合中的应用是有效及可行的。     

Cubic spline curve fitting with controlled end conditions

Several approximate methods for cubic spline curve fitting have been developed and successfully used. This paper presents a more flexible version of a proven technique by using a set of end conditions suggested by Nutbourne. The advantages and disadvantages of several techniques are clarified and sample graphical output is given. The results should be of greatest interest to users of inexpensive, computer graphics equipment who are interested in improving passive graphical output.