2013, 3(3): 471-489. doi: 10.3934/naco.2013.3.471

Introduction to the theory of splines with an optimal mesh. Linear Chebyshev splines and applications

1. 

Prof. N. E. Zhukovsky Central Aerohydrodynamic Institute (TsAGI), Zhukovsky str., 1, Zhukovsky, Moscow region, 140180, Russian Federation

2. 

Moscow Institute of Physics and Technology (State University) (MIPT), Institutsky Lane 9, Dolgoprudny, Moscow region, 141700, Russian Federation

Received  February 2012 Revised  July 2013 Published  July 2013

On June 18, 2008 at the Plenary Meeting of the International Conference ``Differential Equations and Topology" dedicated to the 100-th anniversary of L.S. Pontryagin, the report [15] was submitted by Isaev V.K. and Leitmann G. This report in a summary form included a section dedicated to the research of scientists of TsAGI in the field of automation of full life-cycle (i.e. engineering-design-manufacturing, or CAE/CAD/CAM, or CALS-technologies) of wind tunnel models [21]. Within this framework, methods of geometric modeling [1,11] were intensively developed, new classes of optimal splines have been built, including the Pontryagin splines and the Chebyshev splines [12-13,19,37]. This paper reviews some results on the Chebyshev splines. We also give brief remarks about the new applications of Chebyshev splines (outside the usual scope of CALS-technologies in design and manufacturing), namely to the actual problem of air traffic management (ATM) within the Free Flight concept.
Citation: Vyacheslav K. Isaev, Vyacheslav V. Zolotukhin. Introduction to the theory of splines with an optimal mesh. Linear Chebyshev splines and applications. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 471-489. doi: 10.3934/naco.2013.3.471
References:
[1]

G. A. Amir'yants, N. A. Vladimirova, V. M. Gadetskiy, V. K. Isaev and S. V. Skorodumov, The nonlinear problems of integrated aerodynamic modelling,, Nonlinear dynamic analysis (NDA2), (2002). Google Scholar

[2]

D. L. Barrow, C. K. Chui, P. W. Smith and J. D. Ward, Unicity of best approximation by second order splines with variable knots,, Mathematics of Computation, 32 (1978), 11. Google Scholar

[3]

P. L. Chebyshev, Questions about the least quantities related to the approximate representation of functions,, Full. Works, 2 (1948). Google Scholar

[4]

P. L. Chebyshev, The theory of mechanisms known as parallelograms,, Full. Works, 2 (1948). Google Scholar

[5]

M. G. Cox, An algorithm for approximating convex functions by means of first - degree splines,, Computer J., 14 (1971). Google Scholar

[6]

V. K. Dzyadyk, "Introduction to the Theory of Uniform Approximation of Functions by Polynomials,", Nauka, (1977). Google Scholar

[7]

E. A. Fedosov, The programs of development of systems of air traffic management in Europe and the U. S. SESAR and NextGen (Analytical review of the materials of foreign sources of information),, General Editor (O. V. Degtyaryov and I. F. Zubkova compilers), (2011). Google Scholar

[8]

V. V. Filatov, On Chebyshev approximation by cubic splines,, Computer Systems, (1973). Google Scholar

[9]

A. I. Grebennikov, "The Method of Splines and Solving Ill-posed Problems in Approximation Theory,", Lomonosov MSU Press, (1983). Google Scholar

[10]

K. Ichida and T. Kiuopo, Segmentation of planar curve,, Electronics and Communication in Japan, 58-d (1975). Google Scholar

[11]

V. K. Isaev, "Geometrical Fundamentals of the CAE/CAD/CAM-system for Wind Tunnel Models,", Doctoral Dissertation, (1991). Google Scholar

[12]

V. K. Isaev, Pontryagin maximum principle and controlled processes of Hermitian interpolation,, Modern problems of mathematics, 167 (1985), 156. Google Scholar

[13]

V. K. Isaev, To the theory of optimal splines,, Applied Mathematics and Computation (Special Issue in Honor of George Leitmann on his 86th Birth year), 217 (2010), 1095. doi: 10.1016/j.amc.2010.05.051. Google Scholar

[14]

V. K. Isaev, B. Kh. Davidson, E. N. Khobotov and V. V. Zolotukhin, On construction of multi-level intellectual air traffic management system,, Proceedings of the Third International Conference, I (2009), 5. Google Scholar

[15]

V. K. Isaev and G. Leitmann, Brief comments on the half-centennial history (1957-2007),, Differential Equations and Topology: International conference dedicated to the Centennial Anniversary of L. S. Pontryagin (1908-1988), (2008), 1908. Google Scholar

[16]

V. K. Isaev and S. A. Plotnikov, On approximation of functions by splines of the first degree, Methods of spline functions in numerical analysis (Computer Systems),, Mathem. Institute, (1983), 27. Google Scholar

[17]

V. K. Isaev and S. A. Plotnikov, The algorithm of polygonal approximation to a given accuracy and a minimal number of nodes, Recent advances in the machining of curved surfaces on CNC machines,, LDNTP Press, (1983), 42. Google Scholar

[18]

V. K. Isaev and S. A. Plotnikov, The inverse problem of optimal Chebyshev approximation of geometric information,, Trudy TsAGI, 2344 (1987), 3. Google Scholar

[19]

V. K. Isaev and S. A. Plotnikov, The reverse Chebyshev problem and Chebyshev splines,, Optimal control and differential equations: To the seventieth anniversary from the day of birth of academician E. F. Mishchenko, (1995), 164. Google Scholar

[20]

V. K. Isaev, S. A. Plotnikov, V. P. Sitnikov and N. V. Shcherbakov, "Some Problems of Optimization of Trajectories Machining Parts with Complex Technical Forms,", Experience and prospects for effective use of technological equipment with CNC, (1982). Google Scholar

[21]

V. K. Isaev, V. P. Sitnikov, V. A. Sukhnev, I. G. Karimullin, S. V. Skorodumov, V. V. Sonin, V. V. Lubashevskiy, O. E. Baryshnikov, V. E. Zaytsev, E. N. Khobotov, L. I. Shustova and V. M. Platov, Research on creation of the CAE/CAD/CAM-system for wind tunnel models in TsAGI: ASIM (1970-1980), ASIM+ (1980-1992),, Problems of creation of perspective air-space technique, (2005), 498. Google Scholar

[22]

V. K. Isaev and V. V. Zolotukhin, Some problems of 2D-maneuvering to ensure the vortex safety of an aircraft,, Aerospace MAI Journal, 16 (2009), 5. Google Scholar

[23]

V. K. Isaev and V. V. Zolotukhin, Construction of plane maneuvers the aircrafts to ensure the air traffic safety,, Proceedings on the VIII Internatonal conference on nonequilibrium processes in nozzles and jets (NPNJ 2010) (May, (2010), 25. Google Scholar

[24]

V. K. Isaev and V. V. Zolotukhin, Construction of plane maneuvers the aircrafts to ensure air traffic safety,, X All-Russian Congress on the fundamental problems of theoretical and applied mechanics. (Nizhny Novgorod, IV (2011), 24. Google Scholar

[25]

V. K. Isaev and V. V. Zolotukhin, Intellect air traffic management system based on the Free Flight concept,, Managing the development of large-scale systems (MLSD'2011): Proceedings of the V International Conference, I (2011), 3. Google Scholar

[26]

V. K. Isaev and V. V. Zolotukhin, The basics of construction a multi-level intellectual air traffic management system based on the concept of free flight,, Proceedings of the XVII International Conference on Computational Mechanics and Modern Applied Software Syste ms (CMMASS'2011), (2011), 25. Google Scholar

[27]

H. M. Johnson and A. A. Uogt, Geometric method for approximating convex arc,, SIAM J. Appl. Math., 38 (1980), 317. doi: 10.1137/0138027. Google Scholar

[28]

Yu. L. Ketkov, On optimal methods of piecewise linear approximation,, Proceedings of the USSR universities, 9 (1966), 1202. Google Scholar

[29]

A. K. Khmelyov, "The Methods of Approximation of Functions and Curves by Splines with a Minimum Number of Nodes and Applications to the problem of design surface of the wind tunnel models,", PhD thesis, (1989). Google Scholar

[30]

Yoshisuke Kirozumi and W. A. Dawis, Poligonal approximation by minimax method,, Computer Graphics and Image Proc., 19 (1982), 248. Google Scholar

[31]

N. P. Korneichuk, "Splines in Approximation Theory,", Nauka, (1984). Google Scholar

[32]

U. Montanari, A note on minimal length polygonal approximation to a digitized contour,, Comm. ACM, 13 (1970), 41. Google Scholar

[33]

T. Pavlidis, Poligonal approximations by Newton's method,, IEEE Trans. Comput., C-26 (1977), 801. Google Scholar

[34]

T. Pavlidis and S. L. Horowitz, Segmentation of plane curves,, IEEE Trans. Comput., C-23 (1974), 860. Google Scholar

[35]

G. M. Phillips, Algorithms for piecewise straight line approximations,, Computer J., 11 (1968), 110. Google Scholar

[36]

S. A. Plotnikov, "Development of Methods for Optimal Approximation of Geometric Information in the CNC Systems,", PhD thesis, (1986). Google Scholar

[37]

S. A. Plotnikov, On the optimal approximation to a given accuracy of the trajectories of discrete control systems,, Depon. VINITI, (): 3690. Google Scholar

[38]

B. A. Popov, The accuracy of approximation by uniform splines (absolute error),, PMI UAS, (1983). Google Scholar

[39]

B. A. Popov, The accuracy of the approximation by uniform splines (weighted error),, PMI UAS, (1983). Google Scholar

[40]

B. A. Popov and G. S. Tesler, Approximation of functions for technical applications,, Nauk. Thought, (1980). Google Scholar

[41]

U. E. Ramer, An iterative procedure for the polygonal approximation of plane curves,, Computer Graphics and Image Proc., 1 (1972), 244. Google Scholar

[42]

Ey. Ya. Remez, "Fundamentals of Numerical Methods of Chebyshev Approximation,", Nauk. Dumka, (1969). Google Scholar

[43]

Ey. Ya. Remez, "General Computational Methods of Chebyshev Approximation,", UAS Publ., (1957). Google Scholar

[44]

Ey. Ya. Remez and Gavrilyuk, Computational design of several approaches to the approximate construction of solutions of Chebyshev problems with nonlinear input parameters,, Ukrain Math.J., 12 (1960). Google Scholar

[45]

B. M. Shumilov, On local approximation by splines of first-degree,, Methods of Spline Functions (Computing systems), 75 (1978), 16. Google Scholar

[46]

H. A. Simon, Rational choice and the structure of the environment,, Psychological Review, 63 (1956), 129. Google Scholar

[47]

J. Sklansky, R. L. Chazin and B. J. Hansen, Minimum perimeter polygons of digitized silhuettes,, IEEE Trans. Comput., C.-21 (1972), 445. Google Scholar

[48]

J. Sklansky and V. Gonzales, "Fast Polygonal Approximation of Digitized Curves,", PRIP Proceed., (1979). Google Scholar

[49]

W. C. Stirling, "Satisficing Games and Decision Making: With Applications to Engineering and Computer Science,", Cambridge University Press, (2003). doi: 10.1017/CBO9780511543456. Google Scholar

[50]

I. Tomek, Two algorithms for piecewise liner continuous approximations of functions of one variable,, IEEE Trans. Comput., C-23 (1974), 445. Google Scholar

[51]

H. Werner, "An Introduction to Nonlinear Splines,", Proc. of NATO Advanced Study Institute, (1979). Google Scholar

[52]

C. M. Williams, An efficient algorithm for the piecewse linear approximation of planar curves,, Computer Graphics and Image Proc., 8 (1978), 286. Google Scholar

[53]

Yu. S. Zav'yalov, B. I. Kvasov and V. L. Miroshnichenko, "Methods of Spline Functions,", Nauka, (1980). Google Scholar

[54]

Yu. S. Zav'yalov, V. A. Leus and V. A. Skorospelov, "Splines in Engineering Geometry,", Mashinostroenie, (1985). Google Scholar

[55]

V. V. Zolotukhin, Simulation of vortex wakes in the problems of air traffic control,, Software and Systems, 1 (2011), 126. Google Scholar

[56]

V. V. Zolotukhin, V. K. Isaev and B. Kh. Davidson, Some relevant problems of air traffic management,, Proceedings of MIPT, 1 (2009), 94. Google Scholar

[57]

V. V. Zolotukhin and V. K. Isaev, Application of the satisficing game theory to construct a system to ensure air traffic safety,, Proceedings of the Russian scientific-technical seminar, (2011), 22. Google Scholar

[58]

V. V. Zolotukhin and V. K. Isaev, Methods and models of air traffic management,, Problems of Mechanical Engineering, (2008), 231. Google Scholar

[59]

V. V. Zolotukhin and V. K. Isaev, Using the theory of coalitional games to avoid conflicts between aircrafts,, Proceedings of the 53rd MIPT conference, 2 (2010), 78. Google Scholar

show all references

References:
[1]

G. A. Amir'yants, N. A. Vladimirova, V. M. Gadetskiy, V. K. Isaev and S. V. Skorodumov, The nonlinear problems of integrated aerodynamic modelling,, Nonlinear dynamic analysis (NDA2), (2002). Google Scholar

[2]

D. L. Barrow, C. K. Chui, P. W. Smith and J. D. Ward, Unicity of best approximation by second order splines with variable knots,, Mathematics of Computation, 32 (1978), 11. Google Scholar

[3]

P. L. Chebyshev, Questions about the least quantities related to the approximate representation of functions,, Full. Works, 2 (1948). Google Scholar

[4]

P. L. Chebyshev, The theory of mechanisms known as parallelograms,, Full. Works, 2 (1948). Google Scholar

[5]

M. G. Cox, An algorithm for approximating convex functions by means of first - degree splines,, Computer J., 14 (1971). Google Scholar

[6]

V. K. Dzyadyk, "Introduction to the Theory of Uniform Approximation of Functions by Polynomials,", Nauka, (1977). Google Scholar

[7]

E. A. Fedosov, The programs of development of systems of air traffic management in Europe and the U. S. SESAR and NextGen (Analytical review of the materials of foreign sources of information),, General Editor (O. V. Degtyaryov and I. F. Zubkova compilers), (2011). Google Scholar

[8]

V. V. Filatov, On Chebyshev approximation by cubic splines,, Computer Systems, (1973). Google Scholar

[9]

A. I. Grebennikov, "The Method of Splines and Solving Ill-posed Problems in Approximation Theory,", Lomonosov MSU Press, (1983). Google Scholar

[10]

K. Ichida and T. Kiuopo, Segmentation of planar curve,, Electronics and Communication in Japan, 58-d (1975). Google Scholar

[11]

V. K. Isaev, "Geometrical Fundamentals of the CAE/CAD/CAM-system for Wind Tunnel Models,", Doctoral Dissertation, (1991). Google Scholar

[12]

V. K. Isaev, Pontryagin maximum principle and controlled processes of Hermitian interpolation,, Modern problems of mathematics, 167 (1985), 156. Google Scholar

[13]

V. K. Isaev, To the theory of optimal splines,, Applied Mathematics and Computation (Special Issue in Honor of George Leitmann on his 86th Birth year), 217 (2010), 1095. doi: 10.1016/j.amc.2010.05.051. Google Scholar

[14]

V. K. Isaev, B. Kh. Davidson, E. N. Khobotov and V. V. Zolotukhin, On construction of multi-level intellectual air traffic management system,, Proceedings of the Third International Conference, I (2009), 5. Google Scholar

[15]

V. K. Isaev and G. Leitmann, Brief comments on the half-centennial history (1957-2007),, Differential Equations and Topology: International conference dedicated to the Centennial Anniversary of L. S. Pontryagin (1908-1988), (2008), 1908. Google Scholar

[16]

V. K. Isaev and S. A. Plotnikov, On approximation of functions by splines of the first degree, Methods of spline functions in numerical analysis (Computer Systems),, Mathem. Institute, (1983), 27. Google Scholar

[17]

V. K. Isaev and S. A. Plotnikov, The algorithm of polygonal approximation to a given accuracy and a minimal number of nodes, Recent advances in the machining of curved surfaces on CNC machines,, LDNTP Press, (1983), 42. Google Scholar

[18]

V. K. Isaev and S. A. Plotnikov, The inverse problem of optimal Chebyshev approximation of geometric information,, Trudy TsAGI, 2344 (1987), 3. Google Scholar

[19]

V. K. Isaev and S. A. Plotnikov, The reverse Chebyshev problem and Chebyshev splines,, Optimal control and differential equations: To the seventieth anniversary from the day of birth of academician E. F. Mishchenko, (1995), 164. Google Scholar

[20]

V. K. Isaev, S. A. Plotnikov, V. P. Sitnikov and N. V. Shcherbakov, "Some Problems of Optimization of Trajectories Machining Parts with Complex Technical Forms,", Experience and prospects for effective use of technological equipment with CNC, (1982). Google Scholar

[21]

V. K. Isaev, V. P. Sitnikov, V. A. Sukhnev, I. G. Karimullin, S. V. Skorodumov, V. V. Sonin, V. V. Lubashevskiy, O. E. Baryshnikov, V. E. Zaytsev, E. N. Khobotov, L. I. Shustova and V. M. Platov, Research on creation of the CAE/CAD/CAM-system for wind tunnel models in TsAGI: ASIM (1970-1980), ASIM+ (1980-1992),, Problems of creation of perspective air-space technique, (2005), 498. Google Scholar

[22]

V. K. Isaev and V. V. Zolotukhin, Some problems of 2D-maneuvering to ensure the vortex safety of an aircraft,, Aerospace MAI Journal, 16 (2009), 5. Google Scholar

[23]

V. K. Isaev and V. V. Zolotukhin, Construction of plane maneuvers the aircrafts to ensure the air traffic safety,, Proceedings on the VIII Internatonal conference on nonequilibrium processes in nozzles and jets (NPNJ 2010) (May, (2010), 25. Google Scholar

[24]

V. K. Isaev and V. V. Zolotukhin, Construction of plane maneuvers the aircrafts to ensure air traffic safety,, X All-Russian Congress on the fundamental problems of theoretical and applied mechanics. (Nizhny Novgorod, IV (2011), 24. Google Scholar

[25]

V. K. Isaev and V. V. Zolotukhin, Intellect air traffic management system based on the Free Flight concept,, Managing the development of large-scale systems (MLSD'2011): Proceedings of the V International Conference, I (2011), 3. Google Scholar

[26]

V. K. Isaev and V. V. Zolotukhin, The basics of construction a multi-level intellectual air traffic management system based on the concept of free flight,, Proceedings of the XVII International Conference on Computational Mechanics and Modern Applied Software Syste ms (CMMASS'2011), (2011), 25. Google Scholar

[27]

H. M. Johnson and A. A. Uogt, Geometric method for approximating convex arc,, SIAM J. Appl. Math., 38 (1980), 317. doi: 10.1137/0138027. Google Scholar

[28]

Yu. L. Ketkov, On optimal methods of piecewise linear approximation,, Proceedings of the USSR universities, 9 (1966), 1202. Google Scholar

[29]

A. K. Khmelyov, "The Methods of Approximation of Functions and Curves by Splines with a Minimum Number of Nodes and Applications to the problem of design surface of the wind tunnel models,", PhD thesis, (1989). Google Scholar

[30]

Yoshisuke Kirozumi and W. A. Dawis, Poligonal approximation by minimax method,, Computer Graphics and Image Proc., 19 (1982), 248. Google Scholar

[31]

N. P. Korneichuk, "Splines in Approximation Theory,", Nauka, (1984). Google Scholar

[32]

U. Montanari, A note on minimal length polygonal approximation to a digitized contour,, Comm. ACM, 13 (1970), 41. Google Scholar

[33]

T. Pavlidis, Poligonal approximations by Newton's method,, IEEE Trans. Comput., C-26 (1977), 801. Google Scholar

[34]

T. Pavlidis and S. L. Horowitz, Segmentation of plane curves,, IEEE Trans. Comput., C-23 (1974), 860. Google Scholar

[35]

G. M. Phillips, Algorithms for piecewise straight line approximations,, Computer J., 11 (1968), 110. Google Scholar

[36]

S. A. Plotnikov, "Development of Methods for Optimal Approximation of Geometric Information in the CNC Systems,", PhD thesis, (1986). Google Scholar

[37]

S. A. Plotnikov, On the optimal approximation to a given accuracy of the trajectories of discrete control systems,, Depon. VINITI, (): 3690. Google Scholar

[38]

B. A. Popov, The accuracy of approximation by uniform splines (absolute error),, PMI UAS, (1983). Google Scholar

[39]

B. A. Popov, The accuracy of the approximation by uniform splines (weighted error),, PMI UAS, (1983). Google Scholar

[40]

B. A. Popov and G. S. Tesler, Approximation of functions for technical applications,, Nauk. Thought, (1980). Google Scholar

[41]

U. E. Ramer, An iterative procedure for the polygonal approximation of plane curves,, Computer Graphics and Image Proc., 1 (1972), 244. Google Scholar

[42]

Ey. Ya. Remez, "Fundamentals of Numerical Methods of Chebyshev Approximation,", Nauk. Dumka, (1969). Google Scholar

[43]

Ey. Ya. Remez, "General Computational Methods of Chebyshev Approximation,", UAS Publ., (1957). Google Scholar

[44]

Ey. Ya. Remez and Gavrilyuk, Computational design of several approaches to the approximate construction of solutions of Chebyshev problems with nonlinear input parameters,, Ukrain Math.J., 12 (1960). Google Scholar

[45]

B. M. Shumilov, On local approximation by splines of first-degree,, Methods of Spline Functions (Computing systems), 75 (1978), 16. Google Scholar

[46]

H. A. Simon, Rational choice and the structure of the environment,, Psychological Review, 63 (1956), 129. Google Scholar

[47]

J. Sklansky, R. L. Chazin and B. J. Hansen, Minimum perimeter polygons of digitized silhuettes,, IEEE Trans. Comput., C.-21 (1972), 445. Google Scholar

[48]

J. Sklansky and V. Gonzales, "Fast Polygonal Approximation of Digitized Curves,", PRIP Proceed., (1979). Google Scholar

[49]

W. C. Stirling, "Satisficing Games and Decision Making: With Applications to Engineering and Computer Science,", Cambridge University Press, (2003). doi: 10.1017/CBO9780511543456. Google Scholar

[50]

I. Tomek, Two algorithms for piecewise liner continuous approximations of functions of one variable,, IEEE Trans. Comput., C-23 (1974), 445. Google Scholar

[51]

H. Werner, "An Introduction to Nonlinear Splines,", Proc. of NATO Advanced Study Institute, (1979). Google Scholar

[52]

C. M. Williams, An efficient algorithm for the piecewse linear approximation of planar curves,, Computer Graphics and Image Proc., 8 (1978), 286. Google Scholar

[53]

Yu. S. Zav'yalov, B. I. Kvasov and V. L. Miroshnichenko, "Methods of Spline Functions,", Nauka, (1980). Google Scholar

[54]

Yu. S. Zav'yalov, V. A. Leus and V. A. Skorospelov, "Splines in Engineering Geometry,", Mashinostroenie, (1985). Google Scholar

[55]

V. V. Zolotukhin, Simulation of vortex wakes in the problems of air traffic control,, Software and Systems, 1 (2011), 126. Google Scholar

[56]

V. V. Zolotukhin, V. K. Isaev and B. Kh. Davidson, Some relevant problems of air traffic management,, Proceedings of MIPT, 1 (2009), 94. Google Scholar

[57]

V. V. Zolotukhin and V. K. Isaev, Application of the satisficing game theory to construct a system to ensure air traffic safety,, Proceedings of the Russian scientific-technical seminar, (2011), 22. Google Scholar

[58]

V. V. Zolotukhin and V. K. Isaev, Methods and models of air traffic management,, Problems of Mechanical Engineering, (2008), 231. Google Scholar

[59]

V. V. Zolotukhin and V. K. Isaev, Using the theory of coalitional games to avoid conflicts between aircrafts,, Proceedings of the 53rd MIPT conference, 2 (2010), 78. Google Scholar

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