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Journal articles on the topic 'Splines'
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1
Morozov, Alexander, Gennady Fedotov, Lilia Fedorova, Damir Musharapov, and Lilia Khabieva. "The providing durability of the movable square-sided spline joints by electromechanical treatment of the working surfaces." MATEC Web of Conferences 298 (2019): 00117. http://dx.doi.org/10.1051/matecconf/201929800117.
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The article deals with the operating conditions and causes of loss of performance of movable involute splines joints. In order to ensure the durability of the movable involute splines joints, an effective method of hardening the working surfaces of the splined hub by electromechanical treatment is proposed. The influence of the current on the change of microstructure and microhardness in the electromechanical processing of the working surfaces of the spline bushing was determined. The obtained results of the comparative wear testing of samples involute splines joints depending on the time of testing. It is established that the use of electromechanical processing of the working surfaces of involute splines joints allows reducing the time of their running-in and increasing the wear resistance of the movable joints up to 2 ... 2.5 times.
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Zhao, Guang, Xiangyang Zhao, Liting Qian, Yunbo Yuan, Song Ma, and Mei Guo. "A Review of Aviation Spline Research." Lubricants 11, no. 1 (December 25, 2022): 6. http://dx.doi.org/10.3390/lubricants11010006.
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Splines are irreplaceable in high-speed aviation fields due to their simplicity, reliability, and high specific power. Aviation splines are not only subjected to severe operating mechanical loads, but also sometimes operate under grease-lubricated and non-lubricated environments. All of this results in aviation splines suffering widespread failures. Since the 1960s, many researchers have carried out much research on aviation splines. The wide range of research topics demonstrates the technical challenges of understanding aviation spline. This paper reviews the research of aviation spline from the aspects of failure form, fatigue strength, surface contact stress, effects of lubrication, and misalignment on wear, as well as experiments. Relevant research shows crowned splines can mitigate the spline wear process induced by angular misalignment, and oil-lubricating splines experience almost no wear. This paper also looks forward to the future development directions of aviation splines.
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Barka, Noureddine, Sasan Sattarpanah Karganroudi, Rachid Fakir, Patrick Thibeault, and Vincent Blériot Feujofack Kemda. "Effects of Laser Hardening Process Parameters on Hardness Profile of 4340 Steel Spline—An Experimental Approach." Coatings 10, no. 4 (April 2, 2020): 342. http://dx.doi.org/10.3390/coatings10040342.
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This study displays the effect of laser surface hardening parameters on the hardness profile (case depth) of a splined shaft made of AISI 4340 steel. The approach is mainly based on experimental tests wherein the hardness profile of laser hardened splines is acquired using micro-hardness measurements. These results are then evaluated with statistical analysis (ANOVA) to determine the principal effect and the contributions of each parameter in the laser hardening process. Using empirical correlations, the case depth of splined shaft at tip and root of spline’s teeth is also estimated and verified with measured data. The obtained results were then used to study the sensitivity of the measured case depths according to the evolution of laser process parameters and geometrical factors. The feasibility and efficiency of the proposed approach lead to a reliable statistical model in which the hardness profile of the spline is estimated with respect to its specific geometry.
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Speleers, Hendrik. "Algorithm 1020: Computation of Multi-Degree Tchebycheffian B-Splines." ACM Transactions on Mathematical Software 48, no. 1 (March 31, 2022): 1–31. http://dx.doi.org/10.1145/3478686.
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Multi-degree Tchebycheffian splines are splines with pieces drawn from extended (complete) Tchebycheff spaces, which may differ from interval to interval, and possibly of different dimensions. These are a natural extension of multi-degree polynomial splines. Under quite mild assumptions, they can be represented in terms of a so-called multi-degree Tchebycheffian B-spline (MDTB-spline) basis; such basis possesses all the characterizing properties of the classical polynomial B-spline basis. We present a practical framework to compute MDTB-splines, and provide an object-oriented implementation in Matlab . The implementation supports the construction, differentiation, and visualization of MDTB-splines whose pieces belong to Tchebycheff spaces that are null-spaces of constant-coefficient linear differential operators. The construction relies on an extraction operator that maps local Tchebycheffian Bernstein functions to the MDTB-spline basis of interest.
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Ezhov, Nikolaj, Frank Neitzel, and Svetozar Petrovic. "Spline approximation, Part 1: Basic methodology." Journal of Applied Geodesy 12, no. 2 (April 25, 2018): 139–55. http://dx.doi.org/10.1515/jag-2017-0029.
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Abstract In engineering geodesy point clouds derived from terrestrial laser scanning or from photogrammetric approaches are almost never used as final results. For further processing and analysis a curve or surface approximation with a continuous mathematical function is required. In this paper the approximation of 2D curves by means of splines is treated. Splines offer quite flexible and elegant solutions for interpolation or approximation of “irregularly” distributed data. Depending on the problem they can be expressed as a function or as a set of equations that depend on some parameter. Many different types of splines can be used for spline approximation and all of them have certain advantages and disadvantages depending on the approximation problem. In a series of three articles spline approximation is presented from a geodetic point of view. In this paper (Part 1) the basic methodology of spline approximation is demonstrated using splines constructed from ordinary polynomials and splines constructed from truncated polynomials. In the forthcoming Part 2 the notion of B-spline will be explained in a unique way, namely by using the concept of convex combinations. The numerical stability of all spline approximation approaches as well as the utilization of splines for deformation detection will be investigated on numerical examples in Part 3.
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Hong, J., D. Talbot, and A. Kahraman. "A generalized semi-analytical load distribution model for clearance-fit, major-fit, minor-fit, and mismatched splines." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 230, no. 7-8 (September 7, 2015): 1126–38. http://dx.doi.org/10.1177/0954406215603741.
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A generalized semi-analytical load distribution model for all common types of involute spline joints is proposed. It is formulated to model clearance-fit (side-fit) involute splines with contacts happening only at the drive flanks of the spline teeth, major or minor-diameter fit splines where additional contact occurs along the top land and the root land of the external spline teeth, respectively, as well as mismatched splines where an intentional lead mismatch is introduced to initiate contact along both drive and back (coast) flanks. Using this model, load distribution and tooth-to-tooth load sharing of example major and minor diameter-fit spline joints under typical multi-directional spur and helical gear loading conditions are characterized and compared to those of the corresponding side-fit spline joint. Further, self-centralizing performance of major and minor-fit splines versus side-fit splines is quantified including its sensitivity to the radial clearance magnitude. Finally, load distribution of an example side-fit spline having various intentional mismatch magnitudes at different torque level is investigated to show that a given mismatch value is optimal at a certain design torque.
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Mijiddorj, Renchin-Ochir, and Tugal Zhanlav. "Algorithm to construct integro splines." ANZIAM Journal 63 (November 16, 2021): 359–75. http://dx.doi.org/10.21914/anziamj.v63.15855.
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We study some properties of integro splines. Using these properties, we design an algorithm to construct splines \(S_{m+1}(x)\) of neighbouring degrees to the given spline \(S_{m}(x)\) with degree \(m\). A local integro-sextic spline is constructed with the proposed algorithm. The local integro splines work efficiently, that is, they have low computational complexity, and they are effective for use in real time. The construction of nonlocal integro splines usually leads to solving a system of linear equations with band matrices, which yields high computational costs. doi:10.1017/S1446181121000316
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Kačala, Viliam, and Csaba Török. "Optimal Approximation of Biquartic Polynomials by Bicubic Splines." EPJ Web of Conferences 173 (2018): 03012. http://dx.doi.org/10.1051/epjconf/201817303012.
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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.
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Xue, Xiangzhen, Wei Yu, Kuan Lin, Ning Zhang, Li Xiao, and Yiqiang Jiang. "Design Method and Teeth Contact Simulation of PEEK Involute Spline Couplings." Materials 17, no. 1 (December 22, 2023): 60. http://dx.doi.org/10.3390/ma17010060.
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In order to design an involute spline made of PEEK (polyetheretherketone)-based material with better performance and improve the design rules of involute splines, initially, an involute splines design theory for PEEK (polyetheretherketone)-based materials is presented, which combines international standards (ISO 4156, 1. 2. 3, 2005), American standards (ANSI B92.2M. 1989), and traditional empirical formulas. Second, using the involute splines calibration method in international standards as a guide, we developed the involute splines calibration method for PEEK-based materials by estimating the impact of energy consumption caused by viscoelasticity on temperature field calibration. Next, the contact characteristics of the designed spline were analyzed using ABAQUS2022 software to confirm the accuracy and reliability of the design and calibration methods. Finally, finite element simulation was used to analyze the influence of different pressure angles, moduli, combined lengths, and other parameters on the contact characteristics of the spline in order to realize the optimal design of PEEK-based material involute splines, to offer a theoretical foundation and improved design methodology for cylindrical straight-tooth involute splines.
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Wang, Zhihua, Falai Chen, and Jiansong Deng. "Evaluation Algorithm of PHT-Spline Surfaces." Numerical Mathematics: Theory, Methods and Applications 10, no. 4 (September 12, 2017): 760–74. http://dx.doi.org/10.4208/nmtma.2017.0003.
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AbstractPHT-splines are a type of polynomial splines over hierarchical T-meshes which posses perfect local refinement property. This property makes PHT-splines useful in geometric modeling and iso-geometric analysis. Current implementation of PHT-splines stores the basis functions in Bézier forms, which saves some computational costs but consumes a lot of memories. In this paper, we propose a de Boor like algorithm to evaluate PHT-splines provided that only the information about the control coefficients and the hierarchical mesh structure is given. The basic idea is to represent a PHT-spline locally in a tensor product B-spline, and then apply the de-Boor algorithm to evaluate the PHT-spline at a certain parameter pair. We perform analysis about computational complexity and memory costs. The results show that our algorithm takes about the same order of computational costs while requires much less amount of memory compared with the Bézier representations. We give an example to illustrate the effectiveness of our algorithm.
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Istiqomatul Fajriyah Yuliati and Pardomuan Sihombing. "Pemodelan Fertilitas Di Indonesia Tahun 2017 Menggunakan Pendekatan Regresi Nonparametrik Kernel dan Spline." Jurnal Statistika dan Aplikasinya 4, no. 1 (June 30, 2020): 48–60. http://dx.doi.org/10.21009/jsa.04105.
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Tujuan dari penelitian ini adalah untuk menganalisis pola hubungan Total Fertility Rate (TFR) dengan Contraceptive Prevalence Rate (CPR). Analisis yang sering digunakan untuk pemodelan adalah analisis regresi. Analisis regresi menurut pendekatannya dapat dibedakan menjadi dua, parametrik dan nonparametrik. Metode regresi nonparametrik yang sering digunakan adalah regresi kernel dan spline. Pada penelitian ini untuk regresi kernel yang digunakan adalah regresi kernel dengan metode penaksir Nadaraya-Watson (NWE) dan penaksir polinomial lokal (LPE), sedangkan untuk regresi spline yang digunakan adalah smoothing spline dan b-splines. Hasil pengepasan kurva (fitting curve) menunjukkan bahwa model regresi nonparametrik terbaik adalah model regresi b-splines dengan degree 2 dan jumlah knot 5. Hal ini dikarenakan model regresi b-splines memiliki kurva yang halus dan terlihat lebih mengikuti sebaran data dibandingkan kurva model regresi lainnya. Model regresi b-splines terpilih memiliki nilai koefisien determinasi R2 sebesar 76.86%, artinya besarnya variasi variabel TFR yang dijelaskan oleh model regresi b-splines sebesar 76.86%, sedangkan sisanya 23.14% dijelaskan oleh variabel lainnya yang tidak dimasukkan ke dalam model.
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Ustuner, K. F., and L. A. Ferrari. "Discrete splines and spline filters." IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing 39, no. 7 (July 1992): 417–22. http://dx.doi.org/10.1109/82.160167.
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Zhao, Yanchun, Mengzhu Zhang, Qian Ni, and Xuhui Wang. "Adaptive Nonparametric Density Estimation with B-Spline Bases." Mathematics 11, no. 2 (January 5, 2023): 291. http://dx.doi.org/10.3390/math11020291.
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Learning density estimation is important in probabilistic modeling and reasoning with uncertainty. Since B-spline basis functions are piecewise polynomials with local support, density estimation with B-splines shows its advantages when intensive numerical computations are involved in the subsequent applications. To obtain an optimal local density estimation with B-splines, we need to select the bandwidth (i.e., the distance of two adjacent knots) for uniform B-splines. However, the selection of bandwidth is challenging, and the computation is costly. On the other hand, nonuniform B-splines can improve on the approximation capability of uniform B-splines. Based on this observation, we perform density estimation with nonuniform B-splines. By introducing the error indicator attached to each interval, we propose an adaptive strategy to generate the nonuniform knot vector. The error indicator is an approximation of the information entropy locally, which is closely related to the number of kernels when we construct the nonuniform estimator. The numerical experiments show that, compared with the uniform B-spline, the local density estimation with nonuniform B-splines not only achieves better estimation results but also effectively alleviates the overfitting phenomenon caused by the uniform B-splines. The comparison with the existing estimation procedures, including the state-of-the-art kernel estimators, demonstrates the accuracy of our new method.
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WULANDARY, SEPTIE, and DRAJAT INDRA PURNAMA. "PERBANDINGAN REGRESI NONPARAMETRIK KERNEL DAN B-SPLINES PADA PEMODELAN RATA-RATA LAMA SEKOLAH DAN PENGELUARAN PERKAPITA DI INDONESIA." Jambura Journal of Probability and Statistics 1, no. 2 (November 18, 2020): 89–97. http://dx.doi.org/10.34312/jjps.v1i2.7501.
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Analisis regresi merupakan salah satu alat statistik yang banyak digunakan untuk mengetahui hubungan antara dua variabel acak atau lebih. Metode penaksiran model regresi terbagi atas regresi parametrik dan nonparametrik. Penelitian ini bertujuan menganalisis pola hubungan pengeluaran perkapita terhadap rata-rata lama sekolah di Indonesia tahun 2018 melalui perbandingan regresi nonparametrik, yaitu regresi kernel dan spline. Regresi kernel yang digunakan adalah regresi kernel dengan metode penaksir Nadaraya-Watson (NWE), sedangkan regresi spline yang digunakan adalah B-Splines. Berdasarkan nilai Generalized Cross Validation (GCV) yang minimum dari model regresi B-Splines, digunakan model dengan degree 2. Perbandingan model terbaik antara model NWE dan B-Splines dilakukan berdasarkan nilai RMSE terkecil dan kurva yang dihasilkan. Pada penelitian ini, model yang terbaik adalah model B-Splines karena memiliki RMSE 0,705, lebih kecil dibandingkan NWE dengan RMSE 1,854. Selain itu, regresi B-Splines memiliki kurva yang halus dan mengikuti sebaran data dibandingkan kurva NWE.
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Burova, I. G., and G. O. Alcybeev. "The Application of Splines of the Seventh Order Approximation to the Solution of Integral Fredholm Equations." WSEAS TRANSACTIONS ON MATHEMATICS 22 (May 25, 2023): 409–18. http://dx.doi.org/10.37394/23206.2023.22.48.
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There are various numerical methods for solving integral equations. Among the new numerical methods, methods based on splines and spline wavelets should be noted. Local interpolation splines of a low order of approximation have proved themselves well in solving differential and integral equations. In this paper, we consider the construction of a numerical solution to the Fredholm integral equation of the second kind using spline approximations of the seventh order of approximation. The support of the basis spline of the seventh order of approximation occupies seven grid intervals. We apply various modifications of the basis splines of the seventh order of approximation at the beginning, the middle, and at the end of the integration interval. It is assumed that the solution of the integral equation is sufficiently smooth. The advantages of using splines of the seventh order of approximation include the use of a small number of grid nodes to achieve the required error of approximation. Numerical examples of the application of spline approximations of the seventh order for solving integral equations are given.
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Rahayu, Putri Indi, and Pardomuan Robinson Sihombing. "PENERAPAN REGRESI NONPARAMETRIK KERNEL DAN SPLINE DALAM MEMODELKAN RETURN ON ASSET (ROA) BANK SYARIAH DI INDONESIA." JURNAL MATEMATIKA MURNI DAN TERAPAN EPSILON 14, no. 2 (March 2, 2021): 115. http://dx.doi.org/10.20527/epsilon.v14i2.2968.
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Sharia Bank Return On Assets (ROA) modeling in Indonesia in 2018 aims to analyze the relationship pattern of Retturn On Assets (ROA) with interest rates. The analysis that is often used for modeling is regression analysis. Regression analysis is divided into two, namely parametric and nonparametric. The most commonly used nonparametric regression methods are kernel and spline regression. In this study, the nonparametric regression used was kernel regression with the Nadaraya-Watson (NWE) estimator and Local Polynomial (LPE) estimator, while the spline regression was smoothing spline and B-splines. The fitting curve results show that the best model is the B-splines regression model with a degree of 3 and the number of knots 5. This is because the B-splines regression model has a smooth curve and more closely follows the distribution of data compared to other regression curves. The B-splines regression model has a determination coefficient of R ^ 2 of 74.92%,%, meaning that the amount of variation in the ROA variable described by the B-splines regression model is 74.92%, while the remaining 25.8% is explained by other variables not included in the model.
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Chen, Wenyu, Yusha Li, Jianjiang Pan, Jianmin Zheng, and Yiyu Cai. "Effective T-spline representation for VRML." International Journal of Virtual Reality 11, no. 1 (January 1, 2012): 15–24. http://dx.doi.org/10.20870/ijvr.2012.11.1.2833.
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Less control points are needed to represent a shape in T-Splines compared to NURBS and subsequently less time is spent in modeling. While getting more and more accepted by commercial software, T-splines, however, are yet part of VRML/X3D. The T-spline VRML is proposed in this work. An effective data structure is designed for T-splines to support online visualization. Compared to the NURBS and the polygonal representations, the proposed T-spline data structure representation can significantly reduce the VRML file size which is a central concern in online applications. As such, complex objects modeled in T-spline form have better chances for real-time transfer from servers to clients. Similar to other VRML nodes, T-spline VRML node can support geometry, color and texture. Users can interact with T-spline more effectively for LOD and animation applications.
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Wang, Shenying, Zhichao Hu, Fengwei Gu, Baoliang Peng, Youqing Chen, Feng Wu, and Yongwei Wang. "A Rapid Manufacturing Method for Rectangular Splines Based on Laser Cutting and Welding." Transactions of the ASABE 64, no. 1 (2021): 117–26. http://dx.doi.org/10.13031/trans.14216.
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HighlightsA rapid manufacturing method for internal and external rectangular spline shafts for use in agricultural machinery was developed using a combination of laser cutting and welding.The shear strength of the internal spline welds, extrusion strength of the spline tooth surfaces, and extrusion and shear strength of the external spline pins were tested.Threshold values were obtained for the average diameter of the internal and external splines.Two case studies (light load and heavy load) were performed to verify the feasibility and reliability of the method.Abstract. In recent years, special-sized spline shafts and gears have been widely used in the trial production of new agricultural machinery in China. However, due to the high production cost and long development cycle of these common components, the development of new agricultural machinery has been affected. To solve these problems, this article proposes a method for rapid manufacturing of rectangular internal and external splines using a combination of laser cutting and welding. Through analysis of the weld shear strength of the internal splines, the extrusion strength of the spline tooth surfaces, and the extrusion and shear strength of the external spline pins, it was calculated that the threshold of the average diameter (dm) of the internal splines, commonly used in agricultural machinery, was dm = 31.17 mm, and that of the external splines was dm = 33.45 mm. The feasibility and reliability of the method were verified with two case studies using light and heavy load conditions. The light load case study was the splines of the power input shaft of the pickup platform of a peanut harvester, and the heavy load case study was the splines of the total power input shaft of a peanut no-till planter. The case studies indicated that under the light load conditions (average power of 1.13 kW, average torque of 64.1 N·m, average speed of 168.7 rpm, cumulative working time of 48 h, and harvested area of 46.4 ha) and heavy load conditions (average power of 89.36 kW, average torque of 1029.9 N·m, average speed of 828.6 rpm, cumulative working time of 51.5 h, and planted area of 31.7 ha), no spline failure was observed, and the reliability was 100.0%. This article provides a technical reference for the rapid production of special-sized rectangular splines as single pieces or in small batches for trial production, which requires low processing accuracy, of new agricultural machinery products. Keywords: Agricultural machinery, Laser cutting, Manufacturing method, Rectangular spline, Strength test, Torque.
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Sivak, Roman. "DEFINITION OF KINEMATICS OF DEFORMATION BASED ON SPLINE-APPROXIMATIONS." Vibrations in engineering and technology, no. 2(97) (August 27, 2020): 101–7. http://dx.doi.org/10.37128/2306-8744-2020-2-11.
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It is suggested to use smoothing cubic spline functions to approximate the current functions. The structure of the selected functional provides the minimum curvature of the spline and the smallest deviation of the spline from the function smoothed at the nodes. The necessary correlation between these requirements is provided by the choice of weights. The algorithm for approximating the current function is implemented by moving from a grid created by current lines and auxiliary lines to a rectangular grid. To reduce the effect of random errors of experimental information on the current line, a smoothing algorithm with a given accuracy was used. The current function is interpolated from the physical grid to the calculated one. The flow velocities were approximated to calculate the strain rates. To estimate the accuracy of the calculations, it is proposed to use the deviation from the incompatibility condition. The main disadvantage is that their behavior around a point determines their behavior as a whole. In this regard, other approaches to the approximation free of this deficiency have been developed in recent years. One such approach that has proven itself well in both theoretical and practical applications is the use of so-called splines. Splines are called functions that are glued together from different pieces of polynomials by a fixed system. The simplest example is broken. Splines naturally occur in many mechanical problems. For example, the spline shape has an elastic beam with point loads. Among the splines, polynomial splines glued together from pieces of polynomials play the most important role. The development of the theory of such splines and their popularization were facilitated by I. Schoenberg's work. Polynomial splines are beginning to be used in many application problems related to function approximation.
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Sarfraz, Muhammad, Munaza Ishaq, and Malik Zawwar Hussain. "Shape Designing of Engineering Images Using Rational Spline Interpolation." Advances in Materials Science and Engineering 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/260587.
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In modern days, engineers encounter a remarkable range of different engineering problems like study of structure, structure properties, and designing of different engineering images, for example, automotive images, aerospace industrial images, architectural designs, shipbuilding, and so forth. This paper purposes an interactive curve scheme for designing engineering images. The purposed scheme furnishes object designing not just in the area of engineering, but it is equally useful for other areas including image processing (IP), Computer Graphics (CG), Computer-Aided Engineering (CAE), Computer-Aided Manufacturing (CAM), and Computer-Aided Design (CAD). As a method, a piecewise rational cubic spline interpolant, with four shape parameters, has been purposed. The method provides effective results together with the effects of derivatives and shape parameters on the shape of the curves in a local and global manner. The spline method, due to its most generalized description, recovers various existing rational spline methods and serves as an alternative to various other methods includingv-splines, gamma splines, weighted splines, and beta splines.
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Budakçı, Gülter, and Halil Oruç. "Further Properties of Quantum Spline Spaces." Mathematics 8, no. 10 (October 14, 2020): 1770. http://dx.doi.org/10.3390/math8101770.
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We construct q-B-splines using a new form of truncated power functions. We give basic properties to show that q-B-splines form a basis for quantum spline spaces. On the other hand, we derive algorithmic formulas for 1/q-integration and 1/q-differentiation for q-spline functions. Moreover, we show a way to find the polynomial pieces on each interval of a q-spline function.
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Xie, Jin, and Xiaoyan Liu. "The EH Interpolation Spline and Its Approximation." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/745765.
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A new interpolation spline with two parameters, called EH interpolation spline, is presented in this paper, which is the extension of the standard cubic Hermite interpolation spline, and inherits the same properties of the standard cubic Hermite interpolation spline. Given the fixed interpolation conditions, the shape of the proposed splines can be adjusted by changing the values of the parameters. Also, the introduced spline could approximate to the interpolated function better than the standard cubic Hermite interpolation spline and the quartic Hermite interpolation splines with single parameter by a new algorithm.
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MacCarthy, B. L., and N. D. Burns. "An Evaluation of Spline Functions for use in Cam Design." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 199, no. 3 (July 1985): 239–48. http://dx.doi.org/10.1243/pime_proc_1985_199_118_02.
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This paper shows how spline functions can be employed for kinematic motion specification in cam design. The polynomial spline is introduced as a special case of a continuous piecewise function. Cubic and quintic splines are derived and their properties are discussed in the cam design context. It is shown how standard cam laws can be approximated extremely accurately with a small number of points and appropriate boundary conditions. The modified sinusoidal acceleration cam law is used as an example. The application of quintic splines to non-standard and special motions is discussed. The algebraic and B-spline representations of spline functions are compared. The former is considered preferable in this context and a list of useful algorithms is given. The real power of the spline function, in particular the algebraic quintic spline, is its simplicity, ease of computation and adaptability to non-standard design problems. The use of parametrized, deficient and exponential splines is proposed for specific applications.
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Sihombing, Pardomuan Robinson, and Ade Famalika. "Penerapan Analisis Regresi Nonparametrik dengan Pendekatan Regresi Kernel dan Spline." Jurnal Ekonomi Dan Statistik Indonesia 2, no. 2 (August 29, 2022): 172–81. http://dx.doi.org/10.11594/jesi.02.02.05.
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Penelitian ini bertujuan untuk menerapkan regresi nonparametrik menggunakan regresi kernel dan spline. Regresi kernel menggunakan metode penaksir Nadaraya-Watson (NWE) dan penaksir Polinomial Lokal (LPE), sedangkan untuk regresi spline adalah smoothing spline dan b-splines. Metode ini diterapkan dalam menganalisis pola hubungan Pertumbuhan Produksi Industri (PPI) dan Tingkat Pajak Perusahaan (TPP). Hasil pengepasan kurva (fitting curve) menunjukkan bahwa model regresi nonparametrik terbaik adalah model regresi b-splines dengan degree 2 dan jumlah knot 5. Hal ini dikarenakan model regresi b-splines memiliki kurva yang halus dan terlihat lebih mengikuti sebaran data dibandingkan kurva model regresi lainnya. TPP berpengaruh signifikan negatif terhadap PPI artinya kenaikan TPP akan menurunkan PPI. Oleh sebab itu perlu kebijakan yang komprehensif dalam menerapkan nilai TPP agar tetap dapat meningkatkan produktivitas industri.
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Akhmedov, Yunus, and Kakhramon Sharipov. "Smoothness assurance of the restorable surface of the cones of drill bits by cubic splines." E3S Web of Conferences 486 (2024): 03016. http://dx.doi.org/10.1051/e3sconf/202448603016.
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The article covers the issues of the application of the spline functions in engineering practice, in particular, in the design of cone drill bits. The polynomial splines, which most commonly used in computational methods are also described. It is also expedient to indicate three important advantages of the spline function for describing the design of cone drill bits. Obtaining a unified mathematical equation for the technical surface of the cones of drill bits is a routine work. The construction of a cubic spline causes difficulties due to the determination of the parameters of the spline, which requires the solution of a system of linear algebraic equations. At the same time, the constructed splines provide second-order smoothness in any direction of the surface line. Further splines of the fifth degree which is reduced to solving a complex system of equations have been developed.
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Zhao, Zemin, Shuangshuang Zhou, Qiang Liu, Long Zhang, Bin Shen, and Jiaming Han. "Simulation and Experimental Study of Ultrasonic Vibratory Grinding of Internal Splines." Machines 12, no. 10 (October 16, 2024): 732. http://dx.doi.org/10.3390/machines12100732.
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As an important component of mechanical transmission systems, internal splines are widely used in aerospace, industrial equipment, and other fields. However, internal splines are prone to deformation and shrinkage after heat treatment. At present, most internal splines with a pitch circle diameter greater than φ60 mm can be processed and shaped by ordinary corundum grinding wheels, but there is no effective processing method for the shaping of small- and medium-sized internal splines. This paper establishes a single abrasive material removal model; uses Abaqus to simulate three-body free grinding; and analyzes the effects of abrasive rotation angle, rotation speed, and grinding depth on material removal under different conditions. By comparing the tooth lead deviation and tooth direction deviation before and after internal spline grinding, the experimental results show that after ultrasonic vibration grinding, the internal spline tooth profile deviation is reduced by 41.9%, and the tooth direction deviation is reduced by 44.1%, which provides a new processing method for the deformation recovery of internal splines after heat treatment.
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Guo, Mayi, Wei Wang, Gang Zhao, Xiaoxiao Du, Ran Zhang, and Jiaming Yang. "T-Splines for Isogeometric Analysis of the Large Deformation of Elastoplastic Kirchhoff–Love Shells." Applied Sciences 13, no. 3 (January 29, 2023): 1709. http://dx.doi.org/10.3390/app13031709.
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In this paper, we develop a T-spline-based isogeometric method for the large deformation of Kirchhoff–Love shells considering highly nonlinear elastoplastic materials. The adaptive refinement is implemented, and some relatively complex models are considered by utilizing the superiorities of T-splines. A classical finite strain plastic model combining von Mises yield criteria and the principle of maximum plastic dissipation is carefully explored in the derivation of discrete isogeometric formulations under the total Lagrangian framework. The Bézier extraction scheme is embedded into a unified framework converting T-spline or NURBS models into Bézier meshes for isogeometric analysis. An a posteriori error estimator is established and used to guide the local refinement of T-spline models. Both standard T-splines with T-junctions and unstructured T-splines with extraordinary points are investigated in the examples. The obtained results are compared with existing solutions and those of ABAQUS. The numerical results confirm that the adaptive refinement strategy with T-splines could improve the convergence behaviors when compared with the uniform refinement strategy.
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Sablonniére, Paul. "Positive spline operators and orthogonal splines." Journal of Approximation Theory 52, no. 1 (January 1988): 28–42. http://dx.doi.org/10.1016/0021-9045(88)90035-4.
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He, Min, Wenbo Du, Hongbo Gao, Dongyan Cai, Li Yang, Cui Li, and Yu Wu. "Research on precise measurement method of full parameters of spline interlocks." Journal of Physics: Conference Series 2174, no. 1 (January 1, 2022): 012009. http://dx.doi.org/10.1088/1742-6596/2174/1/012009.
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Abstract This paper mainly introduces the types, characteristics and applications of spline, studies the high-precision measurement technology of different types of spline. Based on gear measuring instrument, optical coordinate measuring machine and coordinate measuring machine, the high-precision full-parameter measurement methods of spline. Developed countries in Europe have spline standards in the 1950s. At present, China has formed a relatively complete system of key and spline linkage standards, but the overall level is still far from developed countries. With the revision of national standards, spline coupling has been applied in more and more fields with the development of industry. It also puts forward higher requirements for high-precision detection methods. Especially for rectangular splines, there is no national calibration specification for the traceability of rectangular splines. We determine the measurement methods and parameters according to the national standards, international standards and the actual production process of the factory. We hope that we can help the factory to improve the precision of splines and prolong their working life.
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Liu, Chaoyang, and Xiaoping Zhou. "Spline Surfaces over Arbitrary Topological Meshes: Theoretical Analysis and Application." Numerical Mathematics: Theory, Methods and Applications 9, no. 3 (July 20, 2016): 383–415. http://dx.doi.org/10.4208/nmtma.2016.y14020.
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AbstractBased on polyhedral splines, some multivariate splines of different orders with given supports over arbitrary topological meshes are developed. Schemes for choosing suitable families of multivariate splines based on pre-given meshes are discussed. Those multivariate splines with inner knots and boundary knots from the related meshes are used to generate rational spline shapes with related control points. Steps for up to C2-surfaces over the meshes are designed. The relationship among the meshes and their knots, the splines and control points is analyzed. To avoid any unexpected discontinuities and get higher smoothness, a heart-repairing technique to adjust inner knots in the multivariate splines is designed.With the theory above, bivariate C1-quadratic splines over rectangular meshes are developed. Those bivariate splines are used to generate rational C1-quadratic surfaces over the meshes with related control points and weights. The properties of the surfaces are analyzed. The boundary curves and the corner points and tangent planes, and smooth connecting conditions of different patches are presented. The C1–continuous connection schemes between two patches of the surfaces are presented.
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Yaseen, Muhammad, Qamar Un Nisa Arif, Reny George, and Sana Khan. "Comparative Numerical Study of Spline-Based Numerical Techniques for Time Fractional Cattaneo Equation in the Sense of Caputo–Fabrizio." Fractal and Fractional 6, no. 2 (January 18, 2022): 50. http://dx.doi.org/10.3390/fractalfract6020050.
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This study focuses on numerically addressing the time fractional Cattaneo equation involving Caputo–Fabrizio derivative using spline-based numerical techniques. The splines used are the cubic B-splines, trigonometric cubic B-splines and extended cubic B-splines. The space derivative is approximated using B-splines basis functions, Caputo–Fabrizio derivative is discretized, using a finite difference approach. The techniques are also put through a stability analysis to verify that the errors do not pile up. The proposed scheme’s convergence analysis is also explored. The key advantage of the schemes is that the approximation solution is produced as a smooth piecewise continuous function, allowing us to approximate a solution at any place in the domain of interest. A numerical study is performed using various splines, and the outcomes are compared to demonstrate the efficiency of the proposed schemes.
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Sana, Madiha, and Muhammad Mustahsan. "Finite Element Approximation of Optimal Control Problem with Weighted Extended B-Splines." Mathematics 7, no. 5 (May 20, 2019): 452. http://dx.doi.org/10.3390/math7050452.
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In this research article, an optimal control problem (OCP) with boundary observations is approximated using finite element method (FEM) with weighted extended B-splines (WEB-splines) as basis functions. This type of OCP has a distinct aspect that the boundary observations are outward normal derivatives of state variables, which decrease the regularity of solution. A meshless FEM is proposed using WEB-splines, defined on the usual grid over the domain, R 2 . The weighted extended B-spline method (WEB method) absorbs the regularity problem as the degree of the B-splines is increased. Convergence analysis is also performed by some numerical examples.
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MASSON, ROLAND. "BIORTHOGONAL SPLINE WAVELETS ON THE INTERVAL FOR THE RESOLUTION OF BOUNDARY PROBLEMS." Mathematical Models and Methods in Applied Sciences 06, no. 06 (September 1996): 749–91. http://dx.doi.org/10.1142/s0218202596000328.
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We construct biorthogonal spline wavelets on the interval for different spline orders. We then discuss the accuracy of the construction and test the wavelet transform. Building the corresponding wavelet basis for [Formula: see text] boundary conditions we use it for the resolution of elliptic boundary problems both with pure Galerkin and Petrov-Galerkin schemes. The results are compared from stability and convergence points of view. The main conclusion is that using the biorthogonal counterpart of spline functions as trial spaces for low order splines provides much more advantages than using spline functions themselves as trial spaces for high order splines.
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Hayotov, A. R., F. A. Nuraliev, and G. Sh Abdullaeva. "The Coeffisients of the Spline Minimizing Semi Norm in K2(P3)." International Journal of Analysis and Applications 23 (January 2, 2025): 7. https://doi.org/10.28924/2291-8639-23-2025-7.
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Our goal is to construct an approximation of the unknown function f by Sobolev’s method, we construct an approximation form of unknown function by interpolation splines minimizing the semi norm in K2(P3) Hilbert space. Explicit formulas for coefficients of the interpolation splines are obtained. The resulting interpolation spline is exact for the hyperbolic functions and constant. In the last section, we obtain several absolute errors graph in interpolating functions with the sixth order algebraic-hyperbolic spline, and we compare absolute errors of cubic spline and algebraic-hyperbolic in interpolating several functions. Numerical results show that the sixth-order spline interpolates the functions with higher accuracy than the cubic spline.
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Pepin, A., S. S. Beauchemin, S. Léger, and N. Beaudoin. "A New Method for High-Degree Spline Interpolation: Proof of Continuity for Piecewise Polynomials." Canadian Mathematical Bulletin 63, no. 3 (December 9, 2019): 655–69. http://dx.doi.org/10.4153/s0008439519000742.
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AbstractEffective and accurate high-degree spline interpolation is still a challenging task in today’s applications. Higher degree spline interpolation is not so commonly used, because it requires the knowledge of higher order derivatives at the nodes of a function on a given mesh.In this article, our goal is to demonstrate the continuity of the piecewise polynomials and their derivatives at the connecting points, obtained with a method initially developed by Beaudoin (1998, 2003) and Beauchemin (2003). This new method, involving the discrete Fourier transform (DFT/FFT), leads to higher degree spline interpolation for equally spaced data on an interval $[0,T]$. To do this, we analyze the singularities that may occur when solving the system of equations that enables the construction of splines of any degree. We also note an important difference between the odd-degree splines and even-degree splines. These results prove that Beaudoin and Beauchemin’s method leads to spline interpolation of any degree and that this new method could eventually be used to improve the accuracy of spline interpolation in traditional problems.
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Burova, I. G. "Fredholm Integral Equation and Splines of the Fifth Order of Approximation." WSEAS TRANSACTIONS ON MATHEMATICS 21 (May 20, 2022): 260–70. http://dx.doi.org/10.37394/23206.2022.21.31.
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This paper considers the numerical solution of the Fredholm integral equation of the second kind using local polynomial splines of the fifth order of approximation and the fourth order of approximation (cubic splines). The basis splines in these cases occupy five and four adjacent grid intervals respectively. Different local spline approximations of the fifth (or fourth) order of approximation are used at the beginning of the integration interval, in the middle of the integration interval, and at the end of the integration interval. The construction of the calculation schemes for solving the Fredholm equation of the second kind with these splines is considered. The results of the numerical experiments on the approximation of functions and on the solution of the Fredholm integral equations are presented. The results of the solution of the integral equation which uses the polynomial splines of the fifth order of approximation are compared with ones obtained with cubic splines and with the application of the Simpson’s method. Note that in order to achieve a given error using the approximation with quadratic splines, a denser grid of nodes is required than when using the approximation with the cubic splines or splines of the fifth order of approximation.
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Burova, I. G., G. O. Alcybeev, and S. A. Schiptcova. "Splines of the Second and Seventh Order Approximation and the Stability of the Solution of the Fredholm Integral Equations of the Second Kind." WSEAS TRANSACTIONS ON MATHEMATICS 23 (November 16, 2023): 1–15. http://dx.doi.org/10.37394/23206.2024.23.1.
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This work is a continuation of a series of works on the use of continuous local polynomial splines for solving interpolation problems and for solving the Fredholm integral equation of the second kind. Here the construction of a numerical solution to the Fredholm integral equation of the second kind using local spline approximations of the second order and the seventh order of approximation is considered. This paper is devoted to the investigation of the stability of the solution of the integral equation using these local splines. Approximation constants are given in the theorem about the error of approximation by the considered splines. Numerical examples of the application of spline approximations of the second and seventh order of approximation for solving integral equations are given.
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Strelkovskaya, Irina, Irina Solovskaya, and Juliya Strelkovska. "Application of real and complex splines in infocommunication problems." Problemi telekomunìkacìj, no. 1(28) (December 22, 2021): 3–19. http://dx.doi.org/10.30837/pt.2021.1.01.
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The work offers the solution to problems of analysis and synthesis of infocommunication systems with the help of real and complex spline functions. The use of the spline approximation method for solving problems of recovery of random signals and self-similar traffic, management of network objects and network as a whole, and procedures of infocommunication objects and networks functioning is offered. To solve the problems of forecasting, in particular, forecasting the characteristics of network traffic and maintaining the QoS characteristics in its service and formation of requirements for network buffer devices, developed spline extrapolation based on different types of real spline functions, namely: linear, quadratic, quadratic B-splines, cubic, cubic B-splines, cubic Hermite splines. As a criterion for choosing the type of spline function, the prediction error is selected, the accuracy of which can be increased by using a particular kind of spline, depending on the object being predicted. The use of complex flat spline functions is considered to solve the class of user positioning problems in the radio access network. In general, the use of real and complex spline functions allows obtaining the results of improving the Quality of Service in the infocommunication network and ensuring the scalability of the obtained solutions.
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Burova, I. G. "The Local Nonpolynomial Splines and Solution of Integro-Differential Equations." WSEAS TRANSACTIONS ON MATHEMATICS 21 (October 24, 2022): 718–30. http://dx.doi.org/10.37394/23206.2022.21.84.
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The application of the local polynomial splines to the solution of integro-differential equations was regarded in the author’s previous papers. In a recent paper, we introduced the application of the local nonpolynomial splines to the solution of integro-differential equations. These splines allow us to approximate functions with a presribed order of approximation. In this paper, we apply the splines to the solution of the integro-differential equations with a smooth kernel. Applying the trigonometric or exponential spline approximations of the fifth order of approximation, we obtain an approximate solution of the integro-differential equation at the set of nodes. The advantages of using such splines include the ability to determine not only the values of the desired function at the grid nodes, but also the first derivative at the grid nodes. The obtained values can be connected by lines using the splines. Thus, after interpolation, we can obtain the value of the solution at any point of the considered interval. Several numerical examples are given.
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Journal, Baghdad Science. "B-splines Algorithms for Solving Fredholm Linear Integro-Differential Equations." Baghdad Science Journal 1, no. 2 (June 6, 2004): 340–46. http://dx.doi.org/10.21123/bsj.1.2.340-346.
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Algorithms using the second order of B -splines [B (x)] and the third order of B -splines [B,3(x)] are derived to solve 1' , 2nd and 3rd linear Fredholm integro-differential equations (F1DEs). These new procedures have all the useful properties of B -spline function and can be used comparatively greater computational ease and efficiency.The results of these algorithms are compared with the cubic spline function.Two numerical examples are given for conciliated the results of this method.
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Zaharov, A., and Y. Zakharova. "Content of the “Geometric Modeling” Course for the “Mathematics and Computer Science” Training Program." Geometry & Graphics 9, no. 4 (April 14, 2022): 35–45. http://dx.doi.org/10.12737/2308-4898-2022-9-4-35-45.
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In this paper has been considered the main content and distinctive features of the “Geometric Modeling” training course for the “Mathematics and Computer Science” training program 02.03.01 (“Mathematical and Computer Modeling” specialization).
The goal of the “Geometric Modeling” course study is the assimilation of mathematical methods for construction of geometric objects with complex curved shapes, and techniques for their computer visualization by using polygons of curves and surfaces. Methods for construction of structures’ curved shapes using spline representations, as well as techniques for construction of surfaces and volumetric geometries using motion operations and basic logical operations on geometric objects are considered. The spline representations include linear and bilinear splines, Hermite cubic splines and Hermite surfaces, natural cubic and bicubic interpolation splines, Bezier curves and surfaces, rational Bezier splines, B-splines and B-spline surfaces, NURBS-curves and NURBS-surfaces, transfinite interpolation methods, and splines of surfaces with triangular form. Logical operations for intersection of two spline curves, and intersection of two parametric surfaces are considered. The principles of scientific visualization and computer animation are considered in this course as well.
Some examples for visualization of initial data and results of curves and surfaces construction in two- and three-dimensional spaces through the software shell developed by authors and used by students while doing tests have been demonstrated. The software shell has a web interface with the WebGL library graphic support. Tasks for four practical studies in a computer classroom, as well as several variations of homework are represented.
The problems occurring in preparation materials for some course sections are discussed, as well as the practical importance of acquired knowledge for the further progress of students.
The paper may be interesting for teachers of “Geometric Modeling” and “Computer Graphics” courses aimed to students with a specialization in mathematics and information, as well as to those who independently develop software interfaces for algorithms of geometric modeling.
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Osborne, M. R., and Tania Prvan. "Smoothness and conditioning in generalised smoothing spline calculations." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 30, no. 1 (July 1988): 43–56. http://dx.doi.org/10.1017/s0334270000006020.
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AbstractWe consider a generalisation of the stochastic formulation of smoothing splines, and discuss the smoothness properties of the resulting conditional expectation (generalised smoothing spline), and the sensitivity of the numerical algorithms. One application is to the calculation of smoothing splines with less than the usual order of continuity at the data points.
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Marinov, Svetlin, Ognyan Alipiev, and Toni Uzunov. "Interference of the profiles when meshing internal straight splines with gear shapers." MATEC Web of Conferences 287 (2019): 01015. http://dx.doi.org/10.1051/matecconf/201928701015.
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The possibilities for cutting internal straight splines with gear shapers by internal meshing, without interference between meshed profiles, are shown in this article. The interference is an undesirable occurrence whereby the straight profile of the splines and the instrumental contour intersect with each other beyond the meshing line. As a result, the shaper teeth cut off a part of the splines’ straight profile and they are made with some defects. The research shows that the undesirable interference depends directly on the parameters of the splined opening and the number of shaper teeth. Also, it is determined that the interference can be avoided by reducing the number of shaper teeth and the height of the splines. Furthermore, the maximum number of shaper teeth for all standard splines for which the interference is absent, are defined in the article. The results are confirmed by computer simulations of the corresponding instrumental meshing.
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Kobasko, Nikolai I. "Quench Process and Steel Chemistry Optimization to Prevent Quench Cracking during Hardening of Splined Semi – Axles." European Journal of Applied Physics 3, no. 2 (April 3, 2021): 23–28. http://dx.doi.org/10.24018/ejphysics.2021.3.2.60.
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In the paper the hardening process of splined semi-axles is discussed and physics of preventing the quench crack formation during intensive quenching (IQ) is explained. It is shown that during IQ process at the splined cylindrical surface very high compressive current band residual stresses are formed which prevent the possibility of quench crack formation on splines. It is enough to optimize the stress distribution through the section of semi-axle and perform IQ process in order to prevent quench cracks formation in splines. It is achieved via optimizing depth of surface hardened layer. In this case the depth of surface hardened layer for cylinders and for cylinders with splines are the same. There is no need to create a special thin shell on splines or perform carburization to create such shell. Due to larger martensite specific volume, it results in surface compressive residual stress formation. Absence of martensite phase at the core eliminates core swelling that could be a reason in tensile surface stresses. The idea is supported by FEM calculations and testing of real semi-axles in industrial condition. The new idea simplifies cardinally technological process and makes it less costly.
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Kahn-Jetter, Zella L., Eugene Hundertmark, and Suzanne Wright. "Comparison of Torque Transmitting Shaft Connectivity Using a Trilobe Polygon Connection and an Involute Spline." Journal of Mechanical Design 122, no. 1 (January 1, 2000): 130–35. http://dx.doi.org/10.1115/1.533556.
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The results of a finite element analysis of a trilobe polygon shaft connection used as an alternative for a spline for torque transmission is presented. These results are compared to the results of a finite element analysis previously performed on an involute spline. It is shown that the tensile stress in the polygon shaft is significantly smaller than in the involute spline and is smaller than all the other stresses in both the shaft and the hub in the polygon connection. Furthermore, the magnitudes and distributions of the maximum principal compressive stress, the shear stress, and the Von Mises stress are nearly the same on the shaft and the hub. It appears that polygonal connections can be more advantageous than splined connections because of lower stresses and the lack of stress concentrations typical of splines. [S1050-0472(00)00601-2]
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Burova, I. G., and G. O. Alcybeev. "Solution of Integral Equations Using Local Splines of the Second Order." WSEAS TRANSACTIONS ON APPLIED AND THEORETICAL MECHANICS 17 (December 31, 2022): 254–58. http://dx.doi.org/10.37394/232011.2022.17.31.
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Splines are an important mathematical tool in Applied and Theoretical Mechanics. Several Problems in Mechanics are modeled with Differential Equations the solution of which demands Finite Elements and Splines. In this paper, we consider the construction of computational schemes for the numerical solution of integral equations of the second kind with a weak singularity. To construct the numerical schemes, local polynomial quadratic spline approximations and second-order nonpolynomial spline approximations are used. The results of the numerical experiments are given. This methodology has many applications in problems in Applied and Theoretical Mechanics
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BEIRÃO DA VEIGA, L., A. BUFFA, G. SANGALLI, and R. VÁZQUEZ. "ANALYSIS-SUITABLE T-SPLINES OF ARBITRARY DEGREE: DEFINITION, LINEAR INDEPENDENCE AND APPROXIMATION PROPERTIES." Mathematical Models and Methods in Applied Sciences 23, no. 11 (July 23, 2013): 1979–2003. http://dx.doi.org/10.1142/s0218202513500231.
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T-splines are an important tool in IGA since they allow local refinement. In this paper we define analysis-suitable T-splines of arbitrary degree and prove fundamental properties: Linear independence of the blending functions and optimal approximation properties of the associated T-spline space. These are corollaries of our main result: A T-mesh is analysis-suitable if and only if it is dual-compatible. Indeed, dual compatibility is a concept already defined and used in L. Beirão da Veiga et al.5 Analysis-suitable T-splines are dual-compatible which allows for a straightforward construction of a dual basis.
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Dokken, T., V. Skytt, and O. Barrowclough. "LOCALLY REFINED SPLINES REPRESENTATION FOR GEOSPATIAL BIG DATA." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XL-3/W3 (August 20, 2015): 565–70. http://dx.doi.org/10.5194/isprsarchives-xl-3-w3-565-2015.
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When viewed from distance, large parts of the topography of landmasses and the bathymetry of the sea and ocean floor can be regarded as a smooth background with local features. Consequently a digital elevation model combining a compact smooth representation of the background with locally added features has the potential of providing a compact and accurate representation for topography and bathymetry. The recent introduction of Locally Refined B-Splines (LR B-splines) allows the granularity of spline representations to be locally adapted to the complexity of the smooth shape approximated. This allows few degrees of freedom to be used in areas with little variation, while adding extra degrees of freedom in areas in need of more modelling flexibility. In the EU fp7 Integrating Project IQmulus we exploit LR B-splines for approximating large point clouds representing bathymetry of the smooth sea and ocean floor. A drastic reduction is demonstrated in the bulk of the data representation compared to the size of input point clouds. The representation is very well suited for exploiting the power of GPUs for visualization as the spline format is transferred to the GPU and the triangulation needed for the visualization is generated on the GPU according to the viewing parameters. The LR B-splines are interoperable with other elevation model representations such as LIDAR data, raster representations and triangulated irregular networks as these can be used as input to the LR B-spline approximation algorithms. Output to these formats can be generated from the LR B-spline applications according to the resolution criteria required. The spline models are well suited for change detection as new sensor data can efficiently be compared to the compact LR B-spline representation.
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Dube, Mridula, and Reenu Sharma. "Cubic TP B-Spline Curves with a Shape Parameter." International Journal of Engineering Research in Africa 11 (October 2013): 59–72. http://dx.doi.org/10.4028/www.scientific.net/jera.11.59.
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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.
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Ezhov, Nikolaj, Frank Neitzel, and Svetozar Petrovic. "Spline Approximation, Part 2: From Polynomials in the Monomial Basis to B-splines—A Derivation." Mathematics 9, no. 18 (September 8, 2021): 2198. http://dx.doi.org/10.3390/math9182198.
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In a series of three articles, spline approximation is presented from a geodetic point of view. In part 1, an introduction to spline approximation of 2D curves was given and the basic methodology of spline approximation was demonstrated using splines constructed from ordinary polynomials. In this article (part 2), the notion of B-spline is explained by means of the transition from a representation of a polynomial in the monomial basis (ordinary polynomial) to the Lagrangian form, and from it to the Bernstein form, which finally yields the B-spline representation. Moreover, the direct relation between the B-spline parameters and the parameters of a polynomial in the monomial basis is derived. The numerical stability of the spline approximation approaches discussed in part 1 and in this paper, as well as the potential of splines in deformation detection, will be investigated on numerical examples in the forthcoming part 3.