Investigation of the algorithm for constructing smooth spatial curves with the ability to specify the curvature and torsion at the nodal points
DOI:
https://doi.org/10.15587/2312-8372.2017.104775Keywords:
segment defined by two points and two first, second and third derivatives, smoothness of the corresponding degreeAbstract
In the existing methods for describing spline vector-parametric surfaces, a rich arsenal of algorithms has been accumulated. Often the development of hardware equipment outperforms the software. In addition, the state of the art often throws new «challenges»to the developer, for example, the need to obtain a channel surface with special properties, with predefined curvature and torsion, with a certain order of smoothness. This is important in the design of product pipelines, exhaust manifolds of internal combustion engines, etc. Therefore, it makes sense to investigate previously unexplored types of control over the future properties of spline curves, and surfaces based on them. Special polynomial splines of higher degrees (seventh degree) and mathematical apparatus (linear algebra, Cramer method, vector-parametric description of curves) are used in the work.
The formula for calculation of the vector-parametric segment of the seventh degree (on two end points, two first, second and third derivatives in them) is derived. The formula allows more flexible control over the shape of the desired spline, arbitrarily setting the initial data.
The algorithm given in the work allows to answer the «call», giving the designer complete freedom in controlling the geometric properties of the object even at the development stage. The algorithm allows to create better samples of modern technology thanks to the control of curvature and torsion.
References
- Golovanov, N. N. (2002). Geometricheskoe modelirovanie. Moscow: Izdatel'stvo Fiziko-matematicheskoi literatury, 472.
- Rogers, D. F., Adams, J. A. (1989). Mathematical Elements for Computer Graphics. Ed. 2. McGraw-Hill Science, 512.
- Faux, I. D., Pratt, M. J. (1980). Computational Geometry for Design and Manufacture. Ellis Horwood Ltd, 329.
- Zavialov, Yu. S., Kvasov, B. I., Miroshnichenko, V. L. (1982). Metody splain-funktsii. Moscow: Nauka, 352.
- Fu, Y. L., Di, H. T. (2009). Simultaneous Measurement of Torsion and Curvature Using Curvature Fiber Optic Sensor. Key Engineering Materials, 392-394, 448–453. doi:10.4028/www.scientific.net/kem.392-394.448
- Rovenski, V. (2000). Geometry of Curves and Surfaces with MAPLE. Birkhäuser Basel, 310. doi:10.1007/978-1-4612-2128-9
- Pogorelov, A. V. (1983). Geometriia. Moscow: Nauka, Matgiz, 288.
- Heller, H. R. (1975). Internationaler Handel. Physica-Verlag HD, 250. doi:10.1007/978-3-642-93617-3
- Lambek, J. (1971). Torsion Theories, Additive Semantics, and Rings of Quotients. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer, 94. doi:10.1007/bfb0061029
- Yaremenko, N. (2014). Derivation of Field Equations in Space with the Geometric Structure Generated by Metric and Torsion. Journal of Gravity, 2014, 1–13. doi:10.1155/2014/420123
- Vasudevaiah, M., Patturaj, R. (1994). Effect of torsion in a helical pipe flow. International Journal of Mathematics and Mathematical Sciences, 17 (3), 553–560. doi:10.1155/s0161171294000803
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2017 Alexander Kovtun
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.
The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.