A Study on a Curve and Three Equations

Öz

The fixed-width curves, 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, three differential equations obtained by means of a system of differential equations characterizing fixed-width curves are examined. These differential equations are third order, variable coefficient, homogeneous, linear differential equations based on different variables. Approximate solutions of these equations are calculated with two different polynomial approximations and error analysis is made for the solutions. Thus, the best approach method is determined for the most accurate result. Also, these equations can 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 with similar equations. 

Anahtar Kelimeler

• Bernstein polynomial approximation
• Hermite polynomial approximation
• Fixed-width curves
• Frenet frame
• Frenet frame

Bir Eğri ve Üç Denklem Üzerine Bir Çalışma

Öz

Kinematik, mühendislik, sanat, cam dizayn ve geometri gibi birçok alanda çok özel bir yere sahip olan sabit genişlikli eğriler bu başlık altında özel olarak ele alınmıştır. Bu çalışmada sabit genişlikli eğrileri karakterize eden bir diferansiyel denklem sisteminin vasıtasıyla elde edilen üç diferansiyel denklem irdelenmiştir. Bu diferansiyel denklemler farklı değişkenlere bağlı üçüncü mertebeden, değişken katsayılı, homojen, lineer diferansiyel denklemlerdir. Bu denklemlerin farklı iki polinom yaklaşımı ile yaklaşık çözümleri hesaplanıp hata analizleri yapılmıştır. Elde edilen sonuçlar sayısal bir örnek üzerinden analiz edilerek en iyi sonuç veren yaklaşım metodu tespit edilmiştir. Bu denklemler farklı uzaylarda farklı çatılara göre farklı eğri tipleri için de bir karakterizasyon teşkil edebilmektedir. Dolayısıyla bu çalışma sadece bu eğri tipi için değil benzer denklemlerle ifade edilebilen tüm eğrilerin geometrisi için önem arz etmektedir.

Anahtar Kelimeler

• Bernstein polinom yaklaşımı
• Hermite polinom yaklaşımı
• Sabit Genişlikli Eğri
• Frenet Çatısı

Kaynakça

1. [1] Ball, N.H. 1930. On Ovals. American Mathematical Monthly, 27, 348-353.
2. [2] Köse, Ö. 1986. On Space Curves of Constant Breadth. Doğa Turk Journal of Mathematics. 10(1), 11-14.
3. [3] Euler, L. 1778- 1780. De Curvis Trangularibis. Acta Academica Petropoliteanca. 3-30.
4. [4] Reuleaux, F. 1963. The Kinematics of Machinery. Trans. By Kennedy A.B.W. Dover Pub. New York.
5. [5] Mellish, A.P. 1931. Notes of Differential Geometry. Annals of Mathematics, 32, 181-190.
6. [6] Struik, D.J. 1931. Differential Geometry in the Large. Bulletin American Mathematical Society. 37, 49-62.
7. [7] Martini, H., Montejano-Peimbert, L., Oliveros, D. 2019. Bodies of Constant Width: An Introduction to Convex Geometry with Applications, Springer Nature Switzerland AG.
8. [8] Zayas E.E., Cardona S., Jordi L. 2009. Analysis and synthesis of the displacement function of the follower in constant-breadth cam mechanisms. Mechanism and Machine Theory. 44, 1938–1949.

9. [9] Shchekotov, M. 2015. Indoor Localization Methods Based on Wi-Fi Lateration and Signal Strength Data Collection. Conference of Open Innovation Association (Fruct). 186-191.
10. [10] Tripathi, A., Gupta, H., Dutta, T., Mishra, R., Shukla, K., Jit, S. 2018. Coverage and Connectivity in WSNs: A Survey. Research Issues and Challenges, IEEE Access, 6, 26971-26992.
11. [11] Zhu, C., Zheng, C., Shu, L., Han, G. 2012. A Survey on Coverage and Connectivity Issues in Wireless Sensor Networks. Journal Network and Computer Applications. 35, 619-632.
12. [12] Kolaei, A., Rakheja,S., Richard M.J. 2014. 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 46, 46–58.
13. [13] Altunkaya, B., Aksoyak, F.K. 2017. Curves of Constant Breadth According to Darboux frame, Commun. Fac. Sci. Univ. Ank. Series A1, 66(2), 44-52.
14. [14] Kocayiğit, H., Önder, M. 2013. Space Curves of Constant Breadth in Minkowski 3-space, Annali di Matematica Pura ed Applicata. 192, 805–814.
15. [15] Aydın, T.A., Sezer, M. 2018. Hermite Polynomial Approach to Determine Spherical Curves in Euclidean 3-space, New Trends in Mathematical Science, 6(3), 189-199.
16. [16] Aydın, T.A., Sezer, M., Kocayiğit, H. 2018. Bernsteinn Polynomials Approach to Determine Timelike Curves of Constant Breadth in Minkowski 3-space. Communication in Mathematical Modeling and Applications. 3 (2), 9-22.
17. [17] Işık, O.R., Sezer, M., Güney, Z. 2011. A Rational Approximation Based on Bernstein Polynomials for High Order Initial and Boundary Values Problems, Applied Mathematics and Computation, 217, 9438-9450.
18. [18] Bhatti, M.I., Brocken, B. 2007. Solutions of Differential Equations in A Bernstein Polynomial Basis. Journal of Computational and Applied Mathematics. 205, 272-280.