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1 Introduction In engineering geodesy point clouds derived from areal (area-wise, in contrast to point-wise) measurement methods such as terrestrial laser scanning or photogrammetry are almost never used as final result. In computer-aided design (CAD) or deformation analyses a curve or surface approximation with a continuous mathematical function is required. An overview of curve and surface approximation of 3D point clouds, including polynomial curves, Bézier curves, B-Spline curves and NURBS curves, is given by Bureick et al. [ 4 ]. This paper will focus on the

References Anthes, R. A., Rocken, C., and Kuo, Y. H. (2000). Applications of COSMIC to Meteorology and Climate. Terr. Atmos. Oceanic Sci., 11:115-156. Foelsche, U., Kirchengast, G., Steiner, A. K., Kornblueh, L., Manzini, E., and Bengtsson, L. (2008). An observing system simulation experiment for climate monitoring with GNSS radio occultation data: Setup and test bed study. Journal of Geophysical Research, 113. Freeden, W. (1981a). On Approximation by Harmonic Splines. Manuscripta Geodaetica, 6:193-244. Freeden, W. (1981b). On Spherical Spline Interpolation and

[1] Ahlberg J.H., Nilson E.N., Walsh J.L., The Theory of Splines and their Applications, Academic Press, New York-London, 1967 [2] de Boor C., A Practical Guide to Splines, Appl. Math. Sci., 27, Springer, New York, 2001 [3] Schumaker L.L., Spline Functions: Basic Theory, 3rd ed., Cambridge Math. Lib., Cambridge University Press, Cambridge, 2007 [4] Volkov Yu.S., Totally positive matrices in the methods of constructing interpolation splines of odd degree, Siberian Adv. Math., 2005, 15(4), 96–125 [5] Zav’yalov Yu

Computational Methods in Applied Mathematics Vol. 13 (2013), No. 1, pp. 39–54 c© 2013 Institute of Mathematics, NAS of Belarus Doi: 10.1515/cmam-2012-0002 On Cardinal Spline Interpolation Rolf D. Grigorieff Dedicated to F. Tröltzsch on the occasion of his sixtieth birthday Abstract — In the present paper it is shown that the interpolation problem for multiple knot cardinal splines subject to general interpolation conditions has a unique solution with polynomial growth if the data grow correspondingly provided a certain determi- nantal condition is satisfied. An

References [1] N. S. Barnett, S.S. Dragomir, A perturbed trapezoid inequality in terms of the fourth derivative , Korean J. Comput. Appl. Math., vol. 9, no. 1, 2002, 45-60. [2] A. M. Bica, M. Curilă, S. Curilă, Two-point boundary value problems associated to functional differential equations of even order solved by iterated splines , Appl. Numer. Math., vol. 110, 2016, 128–147. [3] A. M. Bica, M. Curilă, S. Curilă, Spline iterative method for pantograph type functional differential equations , in Finite Difference Methods. Theory and Applications, I. Dimov, I

the single point precision [ 7 ], the spatiotemporal collocation is based on point clouds modelled by B-spline surfaces. Freeform curves and surfaces like B-splines are a powerful tool to model arbitrary curves and surfaces, respectively. They have their origin in the automobile industry and are a standard tool in computational geometry [ 6 ], but they also find their way into geodesy. In [ 26 ] B-spline surfaces are used as an alternative for spherical harmonics in order to describe the vertical total electron content of the Earth’s atmosphere. The authors of [ 27

1 Introduction The numerical methods for the Navier–Stokes equations have been extensively studied for decades [ 10 , 6 , 11 , 3 ]. The spline methods provide practical ways to obtain high-order smooth approximations for this problem. For example, Botella [ 1 ] developed a B-spline collocation method, and Lai and his coworkers used bivariate spline functions over arbitrary triangulations [ 16 , 14 ]. The main different features between spline methods and finite element methods are listed in [ 16 ]. The two-level method is an efficient strategy to seek

finite difference method. A remarkable method to solve Burgers’ equation is by using splines. The term ‘splines’ is adopted from the name of a flexible strip of metal used by drafters in olden times to assist in drawing curves. Usage of spline functions has great advantages. They are constructed in such a way that they possess high degree of smoothness even at the juncture between two polynomial pieces, i.e. n th degree spline function is continuous up to ( n - 1) th derivative. They are easy to store, manipulate and evaluate on a digital computer. The matrices

J. Numer. Math., Vol. 16, No. 2, pp. 83–106 (2008) DOI 10.1515/ JNUM.2008.004 c de Gruyter 2008 Nonoverlapping domain decomposition with cross points for orthogonal spline collocation B. BIALECKI and M. DRYJA† Received January 31, 2007 Received in revised form March 13, 2008 Abstract — A nonoverlapping domain decomposition approach with uniform and matching grids is used to define and compute the orthogonal spline collocation solution of the Dirichlet boundary value problem for Poisson’s equation on a square partitioned into four squares. The collocation

References [1] Ahlberg J.H., Nilson E.N., Walsh J.L., The Theory of Splines and Their Applications , Academic press New York–London 1967. [2] Drwal G., Grzymkowski R., Kapusta A., Słota D., Mathematica 4 , Wydawnictwo Pracowni Komputerowej J. Skalmierskiego, Gliwice 2000. [3] Drwal G., Grzymkowski R., Kapusta A., Słota D., Mathematica programowanie i zastosowania , Wydawnictwo Pracowni Komputerowej J. Skalmierskiego, Gliwice 1995. [4] Dryja M., Jankowscy J. i M., Przegląd metod i algorytmów numerycznych Część 2 , WNT, Warszawa 1998. [5] Fortuna Z., Macukow D