The Quasi Parallel Curve of a Space Curve

Year 2022, , 203 - 206, 30.06.2021

Fatma Güler

https://doi.org/10.18466/cbayarfbe.955974

Cited By: 1

Abstract

A parallel or offset curve is defined as a curve whose points are a fixed distance from a given curve. These curves are not parallel transport. Often, it has a more complex mathematical structure than the first curve. Offset curves are important in numerically controlled machining, for example, where a biaxial machine defines the shape of the cut made with a round cutting tool. In this study, the quasi parallel curve is defined with the help of the quasi frame of a given curve. According to all selections of the projection vector, the quasi frame equations of this curve are expressed in terms of the quasi elements of the given curve. The curvatures of the quasi parallel curve were obtained depending on the quasi curvatures of the main curve. The study was supported by examples. The examples confirm that the quasi parallel curve is not parallel transport.

Keywords

Space curves, parallel curves, Frenet frame, q-frame.

References

• [1] Wang, F., Liu, H., 2007. Mannheim partner curves in 3-Euclidean space. Mathematics in Practice and Theory, vol. 37, no. 1, pp. 141-143.
• [2] Liu, H., Wang, F., 2008. Mannheim partner curves in 3-space, Journal of Geometry, vol. 88, no. 1-2, pp. 120-126.
• [3] Ravani, B., Ku, T. S., 1991. Bertrand Offsets of ruled and developable surfaces, Comp. Aided Geom.Design, (23), No. 2.
• [4] R.T. Farouki, 1986. The approximation of non-degenerate offset surfaces, Computer Aided Geometric Design, Volume 3, Issue 1, Pages 15-43, ISSN 0167-8396.
• [5] Orbay, K., Kasap, E., Aydemir, I, 2009 .Mannheim Offsets of Ruled Surfaces, Mathematical Problems in Engineering, Volume, Article ID 160917.
• [6] Kasap E, Yüce S, Kuruoğlu N. 2009. The involute-evolute offsets of ruled surfaces. Iranian J Sci Tech Transaction A; 33: 195-201.
• [7] Güler, F., Bayram, E., & Kasap, E. 2018. Offset surface pencil with a common asymptotic curve. International Journal of Geometric Methods in Modern Physics, 15(11), 1850195.
• [8] Güler, F., 2021. Offset Trajectory planning of Robot end Effector and its the Jerk with Curvature Theory. International Journal of Computational Method. Accepted paper.
• [9] Aldossary, M. T., & Gazwani, M. A.2020. Motion of Parallel Curves and Surfaces in Euclidean 3-Space R3. Geometry-math-journal.ro.
• [10] Bishop, Richard L. 1975. There is more than one way to frame a curve. American Mathematical Monthly : 246-251.
• [11] Dede, M., Ekici, C., & Tozak, H. 2015. Directional tubular surfaces. International Journal of Algebra, 9(12), 527-535.
• [12] Baş, S., & Körpinar, T. 2018. Directional Inextensible Flows of Curves by Quasi Frame. Journal of Advanced Physics, 7(3), 427-429.
• [13] Tozak, H., Dede, M., & Ekici, C. 2020. Translation surfaces according to a new frame. Caspian Journal of Mathematical Sciences (CJMS), 9(1), 56-67.
• [14] Körpınar, T., Demirkol, R. C., Körpınar, Z., & Asil, V. 2021. New magnetic flux flows with Heisenberg ferromagnetic spin of optical quasi velocity magnetic flows with flux density. Revista Mexicana de Física, 67(3 May-Jun), 378-392.
• [15] Yazla, A., & Sariaydin, M. T. 2020. On Surfaces Constructed by Evolution According to Quasi Frame. Facta Universitatis, Series: Mathematics and Informatics, 605-619.
• [16] O’Neill, B., 1966. Elementary differential geometry. Academic Press, New York, 59-231.