On a curve and a system

Ağırman Aydın Tuba; Çayan Seda; Sezer Mehmet; Mağden Abdullah

doi:10.30574/gscarr.2020.5.3.0101

Authors

Ağırman Aydın Tuba Faculty of Education, Bayburt University, Bayburt, Turkey.
Çayan Seda Faculty of Science and Arts, Manisa Celal Bayar University, Manisa, Turkey.
Sezer Mehmet Faculty of Science and Arts, Manisa Celal Bayar University, Manisa, Turkey.
Mağden Abdullah Faculty of Engineering and Natural Sciences, Bursa Technical University, Bursa, Turkey.

DOI:

https://doi.org/10.30574/gscarr.2020.5.3.0101

Keywords:

Curves of Constant Width, Hermite Matrix Collocation Method, Taylor Matrix Collocation Method, Frenet Frame, Reuleaux Triangle

Abstract

Curves of constant width, which have a very special place in many fields such as kinematics, engineering, art, cam design and geometry, are specially discussed under this title. In this study, a system of differential equations characterizing the curves of constant width is examined. This is the system of the first order homogenous differential equations with variable coefficients in the normal form. Approximate solutions of the system, by means of two different polynomial approaches, are calculated and error analysis is made. The obtained results are analyzed on a numerical sample and the best method of approach is determined. This system can also constitute a characterization for different types of curves according to different frames in different spaces. Therefore, this study is important not only for this curve type but also for the geometry of all curves that can be expressed in a similar system.

Metrics

Metrics Loading ...

References

Köse Ö. On space curves of constant breadth. Doğa Turk J Math. 1986; 10:11-14.

Martini H, Montejano-Peimbert L, Oliveros D. Bodies of Constant Width: An Introduction to Convex Geometry with Applications. Springer Nature Switzerland AG; 2019.

Euler L. De curvis trangularibis. Acta Academica Petropoliteanca. 1778- 1780.

Reuleaux F. The Kinematics of Machinery. Trans By Kennedy ABW. New York: Dover Pub; 1963.

Altunkaya B, Aksoyak FK. Curves of constant breadth according to Darboux frame. Commun Fac Sci Univ Ank Series A1. 2017; 66: 44-52.

Ball NH. On Ovals. American Mathematical Monthly. 1930; 27:348-353.

Mellish AP. Notes of differential geometry. Annals of Mathem. 1931; 32:181-190.

Struik DJ. Differential geometry in the large. Bulletin Amer Mathem. 1931; 37:49-62.

Zayas EE, Cardona S, Jordi L. Analysis and synthesis of the displacement function of the follower in constant-breadth cam mechanisms. Mechanism and Machine Theory. 2009; 44:1938–1949.

Shchekotov M. Indoor localization methods based on Wi-Fi lateration and signal strength data collection, Conference of Open Innovation Association (Fruct). 2015; 186-191.

Tripathi A, Gupta H, Dutta T, Mishra R, Shukla K, Jit S. Coverage and connectivity in WSNs: A Survey, research issues and challenges. IEEE Access. 2018; 6:26971-26992.

Zhu C, Zheng C, Shu L, Han G, A survey on coverage and connectivity issues in wireless sensor networks. Journal Network and Computer Applications. 2012; 35:619-632.

Kolaei A, Rakheja S, Richard MJ. Effects of tank cross-section on dynamic fluid slosh loads and roll stability of a partly-filled tank truck. European Journal of Mechanics B/Fluids. 2014; 46:46–58.

Aydın TA, Sezer M. Hermite polynomial approach to determine spherical curves in Euclidean 3-space. New Trends in Math Sci. 2018; 6:189-199.

Kocayiğit H, Önder M. Space curves of constant breadth in Minkowski 3-space. Annali di Matematica Pura ed Applicata, 2013; 192:805–814.