Software implementation of algorithms for the t-spline construction and local refinement

Authors: Fedotov D.Yu.
Published in issue: #1(90)/2024
DOI:
Category: Mathematics | Chapter: Computational Mathematics
Keywords: spline, NURBS, T-spline, algorithm, geometric simulation, computer geometry
Published: 26.02.2024

The paper presents results of development and subsequent implementation of algorithms for the T-splines construction and local refinement based on the T-spline theory. NURBS and T-splines generalizing them were reviewed, mathematical apparatus describing them was analyzed, and mathematical justification was provided for their construction and local refinement algorithms by inserting the new points to improve the local area without losing accuracy. Software implementation was developed in the C++ programming language with results visualization using the OpenGL package; examples of the considered algorithms operation were provided. A T-spline was successfully constructed on the given mesh, and several points were inserted into different areas without compromising accuracy of the initially constructed curved surface.
EDN: XXRBHY

References

[1] Zadorozhnyy A.G., Kiselev D.S. Postroenie splaynov c ispol’zovaniem biblioteki OpenGL. Novosibirsk, NGTU, 2019, 88 p. (In Russ.).

[2] Zav’yalov Yu.S., Kvasov B.I., Miroshnichenko V.L. Metody splayn-funktsiy. Moscow, Nauka, 1980, 352 p. (In Russ.).

[3] Bakenov A.S. T-splines: geometrical flexibility and local modification. Geometric Modeling. Computer Graphics in Education. GraphiCon, 2017, pp. 328–331. (In Russ.). EDN: OUROEF

[4] Rodzhers D., Adams Dzh. Matematicheskie osnovy mashinnoy grafiki. Moscow, Mir, 2001, 604 p. (In Russ.).

[5] Les Piegl, Tiller W. The NURBS Book. New York, Springer, 1996, 646 p.

[6] Golovanov N.N. Geometricheskoe modelirovanie. Moscow, INFRA-M. Kurs, 2016, 400 p. (In Russ.).

[7] Sederberg T.W., Zheng J., Bakenov A., Nasri A. T-Splines and T-NURCCs. ACM Transactions on Graphics, 2003, vol. 22, pp. 477–484.

[8] Bazilevs Y., Calo V.M., Cottrell J.A. et al. Isogeometric analysis using T-splines. Computer Methods in Applied Mechanics and Engineering, 2010, vol. 99, pp. 229–263. http://doi.org/10.1016/j.cma.2009.02.036

[9] Sederberg T.W., Cardon D.L., Finnigan G.T. et al. T-spline Simplification and Local Refinement. ACM Transactions on Graphics, 2004, vol. 23, pp. 276–283.

[10] Cardon D.L. T-Spline Simplification. Th. Diss. Brigham Young University, 2007, 114 p.