SPLINE INTERPOLATION WITH THIRD-DEGREE BÉZIER FUNCTIONS

Nicolae URSU-FISCHER, Diana Ioana POPESCU, Ioan RADU

Abstract


The Bézier, B-spline and NURBS curves and surfaces were extensively studied in the literature and used in the shapes design of different products, initially for cars. From the mathematical point of view, in most cases the problems linked to interpolations and approximations with curves and surfaces have been solved. 

Our research team has studied and solved the spline interpolation problem using third-degree Bézier curves between the interpolation points. An efficient algorithm has been set up following the mathematical model which solves the problem. It has been programmed in the C language and used to solve different numerical examples, with results illustrated by diagrams.

It can be observed that there exist multiple variants of interpolation curves due to the imposed interpolation problem specificity: there exist more unknowns than equations (possible conditions to be imposed), that’s why one has to start with some initial values for a pair of selected unknowns.


Full Text:

PDF

References


Barnhill, R., Riesenfeld, R. (ed.), Computer Aided Geometric Design, Academic Press, 1974

Bender, M., Brill, M., Computergrafik. Ein anwendungsorientiertes Lehrbuch, Carl Hansen Verlag, München, 2003, 516 s., ISBN 3-446-22150-6.

Bernstein, S. N., Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Сообщенія Харьковскаго Математическаго Общества, Вторая серія, Томь XIII, 1913, стр. 1-2.

Bézier, P., Mathématiques et CAO. Courbes et surfaces, volume 4, Ed. Hermes, Paris, 1987.

Engeln-Müllges, Gisela, Uhlig, F., Numerical Algorithms with C, Springer, New York, 1996, 596 pp., ISBN 3-540-60530-4.

Farin, Gerald, From conics to NURBS, IEEE Computer Graphics and Applications, vol. 12, no. 5, 1992, pp. 78-86, (NU)

Farin, G. E., Curves and Surfaces for Computer Aided Geometric Design. A Practical Guide, Academic Press, San Diego, sec. ed., 1990, 444 pp., ISBN 0-12-249051-7.

Farin, G., Hoschek, J., Kim, M.-S. (eds.), Handbook of Computer Aided Geometric Design, Elsevier, Amsterdam, 2002, 820 pp., ISBN 0-444-51104-0.

Faux, I. D., Pratt, M. J., Computational Geometry for Design and Manufacture, John Wiley, New York, 1985, 331 pp., ISBN 0-470-27069-1.

Foley, J. D., van Dam, A., Hughes, J. F., Computer Graphics, Principles and Practice, 2/e in C, Addison-Wesley, 1996, 1200 pp.

Forrest, A. R., Interactive interpolation and approximation by Bézier polynomials, Computer Aided Design, 1990, vol. 22, no. 9, 1990, pp. 527-537

Lu, L., Sample-based polynomial appro-ximation of rational Bézier curves, Journal of Computational and Applied Mathematics, 2011, Vol. 235, pp. 1557-1563

Lyche, T., Schumacher, L. L. (ed.), Mathematical Methods in Computer Aided Geometric Design, Academic Press, San Diego, 1989, 611 pp., ISBN 0-12-460515-X

Micula Gh., Functii spline şi aplicaţii, (Spline Functions and Applications – in Romanian), Editura Tehnică, Bucureşti, 1978, 336 pp.

Micula, Gh., Micula Sanda, Handbook of Splines, Springer Science & Business Media, 2012, 606 pp., ISBN 978-94-011-5338-6

Nischwitz, A., Haberäcker, P., Masterkurs Computergrafik und Bildverarbeitung, Friedrich Viewig & Sohn Verlag, Wiesbaden, 2004, 860 s., ISBN 3-528-05874-9.

Popescu, Diana Ioana, Programare în limbajul C (Programming in C- in Romanian), Ed. "DSG Press", Dej, 1999, 288 pp., ISBN 973-98621-4-4.

Popescu, Diana Ioana, Aplicaţii cu SolidWorks. CAD în ingineria mecanică (Applications with SolidWorks. CAD in Mechanical Engineering – in Romanian), Editura Dacia, Cluj-Napoca, 2003, 191 pp., ISBN 973-35-1728-3.

Pozrikidis, C., Numerical Computation in Science and Engineering, University Press, Oxford, 1998, 640 pp., ISBN 0-19-511253-9

Rogers, D. F., Adams, J. A., Mathematical Elements for Computer Graphics, McGraw Hill, Boston, sec. ed., 1990, 611 pp., ISBN 0-07-053530-2.

Стечкин, С. Б., Субботин, Ю. Н., Сплайны в вычислительной математики (Splines in Numerical Computing – in Russian), Наука, Москва, 1976, 248 стр.

Staerk, E., Mehrfach differentzierbare Bézierkurven und Bézierflächen, PhD Thesis, Technische Universität Braun-schweig, 1976.

Stelia O., Potapenko, L., Sirenko, I., On the one method of a third-degree Bézier type spline curve construction, 8 pp., https://arxiv.org/pdf/1712.07485.

Ursu-Fischer, N., Ursu, M., Programare cu C în inginerie (Programming with C in Engineering – in Romanian), Casa Cărţii de Ştiinţă, Cluj-Napoca, 2001, 405 pp., ISBN 973-686-227-5.

Ursu-Fischer, N., Ursu, M., Metode numerice în tehnică şi programe în C/C++ (Numerical Methods in Engineering and Programs in C/C++ - in Romanian), vol. II, Casa Cărţii de Ştiinţă, Cluj-Napoca, 2003, 288 pp., ISBN 973-686-464-2.

Ursu-Fischer, N., Ursu, M., Metode numerice în tehnică (Numerical Methods in Engineering – in Romanian), Casa Cărţii de Ştiinţă, Cluj-Napoca, 2018, approx. 900 pp. (to appear)

Xiao, G., Xu, X., Study on Bézier curve variable step-length algorithm, Physics Procedia, 2012, Vol. 25, pp. 1781-1786.


Refbacks

  • There are currently no refbacks.


JOURNAL INDEXED IN :