Geometric Modeling of Curved Shapes and Cubic Spline Approximation in Engineering Graphics
Keywords:
cubic spline interpolation, geometric modeling, smooth curves, irregular control points, boundary conditionsAbstract
This paper presents a study on cubic spline interpolation, focusing on its application in modeling smooth and continuous curves for engineering graphics. Cubic splines are a series of third-degree polynomial functions connected at control points, ensuring smooth transitions that are critical in fields such as CAD, computer graphics, and animation. This research addresses the challenges encountered when modeling with irregular control points and non-uniform intervals, which can cause traditional cubic spline interpolation methods to generate oscillations or lose continuity. By investigating the mathematical framework of cubic splines, including boundary and continuity conditions, we propose an adaptive approach that improves interpolation performance in these challenging configurations. Computational experiments validate the effectiveness of the method, demonstrating smoother and more accurate curve representations in edge cases. This adaptive cubic spline interpolation approach provides an enhanced solution for geometric modeling, enabling more reliable and visually accurate results in engineering applications.
References
Golovanov, N.N. Geometric modeling [Geometricheskoye modelirovaniye] (in Russian). Physics and Mathematics Literature, 2002.
Djumaniy, J., Yakhshimuratov, K., Khushvakov, S., Savfullaeva, N. Mathematical models and software packages for calculating the balance of information flow, ICISCT 2021.
Djumaniy, J., Khaitov B. Real-time vertical leveling based on GIS and CAD. Journal of Physics: Conference Series, 2022.
Khudaybergenov, T.R., Djumaniy, I.X., et al. Geometric modeling of virtual museum exhibits, 13th International Conference on Smart Cities, 2024.
Barsanti, G., Guidi G. 3D digitization of museum content within the 3dicons project, ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, 2013.
Simeone, G., Bandiera, A. 3D imaging and new ways of making museums interactive, Museologia scientifica, 2016.
Yeisset, P. Reconstruction of volumetric 3D models, 2006.
Franz-Verich, W., Reuter, M., Beinecke, N. Geometric Modeling for Engineering Applications, 2009.
Abdugani Nematov, Turdiev Temur Takhirovich, Ismailov Shikhnazar, Akbar Bakhriddinov. Parallel Computational Algorithms for Solving Boundary Value Problems for Two-Dimensional Equations of Parabolic Type. 2021 International Conference on Information Science and Communications Technologies (ICISCT).
Temur Turdiyev. A one-dimension of nostatsional filtration process of gas in a two-layer poor environment mathematical model and number methods of solving it. Acta of Turin Polytechnic University in Tashkent. 2022/3/31.
TT Turdiyev, B Yu Palvanov, MA Sadikov, KA Salayev, IB Sabirov. Parallel algorithm for the one-dimensional problem of oil movement in a porous medium. Artificial Intelligence, Blockchain, Computing and Security Volume 2. 2023.
Iqbol Madaminovna Karimova, Temur Takhirovich Turdiev. Problems And Major Problems Of Mathematical Modeling Of The Process Of Fluid Movement In Non-Homismal Pool Floors. The American Journal of Engineering and Technology. 2021/2/27.
