On the G-Similarities of two open B-spline curves in R3

Öz

Let G be a transformation group in R3. Any two vectors x and y in R3 are called G-equivalence vectors if there exist a transformation g G such that y = gx satisfies. In this paper the transformation group G will be considered as similarity transformations group or its any subgroup. So if given two vectors x and y in R3 are G-equivalence vectors then these vectors x and y are called G-similar. i.e. rotational, reflectional, translational or scaling similarity. B-spline curves are used basically in Computer Aided Design (CAD), Computer Aided Geometric Design (CAGD), Computer Aided Modeling (CAM). In determining the invariants of spline curves and surfaces at any point, it is necessary to find the analytical equation of each curve and surface and calculate its invariants such as curvature, torsion, principal curvatures, mean and Gaussian curvatures at the desired point. However, it can be very difficult to find the curve or surface to be designed analytically. For example, when a car is designed, the aerodynamic curves in the car will be different from the known surface equation of the car. It is very difficult to write this equation exactly. For these curves and surfaces we designed, the way to overcome this difficulty is to design them with spline curves and surfaces. In this paper the G- equivalence conditions of given two open B-spline curves are studied in case G is similarity transformations group or its any subgroup.

Anahtar Kelimeler

• open B-spline curves
• similarity groups
• G-similar splines

R3 de Açık B-Spline Eğrilerinin G-Benzerlikler

Öz

G, R3 de bir dönüşüm grubu olsun. R3 te herhangi iki x ve y vektörleri verildiğinde eğer bir g G dönüşümü y = gx şartını sağlayacak şekilde bulunabilirse bu iki vektöre G- denk vektörler denir. Bu çalışmada G dönüşüm grubu olarak benzerlik dönüşümleri grubu ve bu grubun tüm altgrupları dikkate alınacaktır. Böylece R3 te herhangi iki x ve y vektörleri G- denk vektörler ise bu vektörlere G-benzer denir. Döndürülme, yansıtılma, ötelenme, ya da germe benzerliği gibi. B-spline eğrileri temelde Bilgisayar Destekli Tasarım (BDT), Bilgisayar Destekli Geometrik Tasarım (BDGT) ya da Bilgisayar Destekli Modelleme (BDM) alanlarında kullanılır. Herhangi bir noktada spline eğri ve yüzeylerinin invaryantlarını belirlemede eğri ve yüzeyin analitik denklemini bulmak ve istenilen noktada eğrilik torsiyon, asal eğrilikler, ortalama ve Gauss eğriliklerini hesaplamak gerekmektedir Oysa ki tasarlanan eğri ve yüzeyde bunu analitik olarak bulmak oldukça zordur. Örneğin, bir araç tasarlandığında, onun aerodinamik yapısından dolayı yüzeyin ve onun üzerindeki eğrilerin analitik denklemini tam olarak bulmak oldukça zordur. Tasarlanan bu eğri ve yüzeyler için bu zorluğun üstesinden gelmenin yolu bunları spline eğri ve yüzeyleri ile tasarlamaktır. Bu çalışmada G, benzerlik dönüşümleri grubu ve onun altgrupları olması durumunda verilen iki B-spline eğrilerinin G- denklik koşulları verilmiştir.

Anahtar Kelimeler

• açık B-spline curves
• benzerlik grupları
• G-benzer splinelar

Kaynakça

1. [1]Incesu M. The Complete System of Point Invariants in the Similarity Geometry, Phd. Thesis, Karadeniz Technical University, Trabzon, 2008.
2. [2]Nagata M., Local Rings, Interscience, NY, 1962.
3. [3]Hadjiev D. Some questions in the theory of vector invariants, Math. USSR-Sbornic. 1 383-396, 1967.
4. [4]Grosshans F. Obsevable groups and Hilbert’s problem, Am. J. Math., 95 229-253, 1973.
5. [5]Bridgman P.W. Dimensional Analysis, 2 Eds., Yale University Press, New Heaven, 1931.
6. [6]Sedov L.I. Similarity and Dimensional Method in Mechanics, English Tr. By V. Kisin, Mir Publishers, USSR, 1982. [7]Langhaar H.L. Dimensional Analysis and Theory of Models, Wiley, 1951.
7. [8]Weyl H. The Classical Groups, Their Invariants and Representations, 2 Eds., with suppl., Princeton, Princeton University Press, 1946.
8. [9]Aripov R.G., Khadzhiev D. The complete system of differential and integral invariants of a curve in Euclidean geometry, Russian Math., 51:7 1-14, 2007.

9. [10]Khadjiev D. An Application of the Invariant Theory to the Differential Geometry of Curves, Fan, Tashkent, 1988.
10. [11]Sagiroglu Y. The equivalence problem for parametric curves in one-dimensional affine space, Int. Math. Forum, 6 177-184, 2011.
11. [12]Sagiroglu Y. Equi-affine differential invariants of a pair of curves, TWMS J. Pure Appl. Math., 6 238-245, 2015.
12. [13]Sagiroglu Y., Peksen O. The equivalence Of Centro-Equiaffine curves, Turk. J. Math., 34 95-104, 2010.
13. [14]Oren I. Complete system of invariants of subspaces of Lorentzian space, Iran. J. Sci. Technol. A., 40 1-8, 2016.
14. [15]Oren I. On invariants of m-vectors in Lorentzian geometry, Int. Electron. J. Geom., 9 38-44, 2016.
15. [16]Khadjiev D., Oren I., Peksen O. Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry, Turk. J. Math., 37 80-94, 2013.
16. [17]Incesu M., Gursoy O. LS (2)-Equivalence conditions of control points and application to planar Bézier curves, New Trends Math. Sci., 5:3 70-84, 2017.
17. [18]Incesu M., Gursoy O. Düzlemsel Bézier Egrilerinin S(2)-Denklik Sartlari, MSU. J. Sci., 5:2 471-477, 2017.
18. [19]Incesu M. The Similarity Orbits in R^ 3, Math: Modelling Application Theory, 2:1 28-37, 2017.
19. [20]İncesu M., Gürsoy O. G= S (1), G= S (2) ve alt Grubları için G-Yörüngeler, MSU. J. Sci., 6:2 595-602, 2018.
20. [21]Ören I., Incesu M. Detecting Similarities of Bézier Curves for the Groups LSim(E2), LSim+(E2) , Conference Proceedings of Science and Technology, 2:2 129-133, 2019.
21. [22]Incesu M. LS (3)-equivalence conditions of control points and application to spatial Bézier curves and surfaces. AIMS Mathematics, 5:2 1216-1246, 2020.
22. [23]Incesu M. R^3 de k- Vektör İçin R(x1,x2,…,xk)^S(3) Cisminin Üreteçleri. International Journal of Advances in Engineering and Pure Sciences., 32:3 239-250, 2020.
23. [24]Farin G. Curvature continuity and offsets for piecewise conics, ACM T. Graphic, 8 89-99, 1989.
24. [25]Farouki R. Exact offsets procedures for simple solids, Comput. Aided. Geom. D., 2 257-279, 1985.
25. [26]Farouki R., Rajan V.T. On the numerical condition of polynomials in Bernstein form, Comput. Aided Geom. D., 4 191-216, 1987.
26. [27]Hoschek J. Offset curves in the plane, Comput. Aided. Des., 17 77-82, 1985.
27. [28]Tiller W., Hanson E. Offsets of two-dimentional profiles, IEEE Comput. Graph., 4 36-46, 1984.
28. [29]Potmann H. Rational curves and surfaces with rational offsets, Comput. Aided. Geom. D., 12 175-192, 1995.
29. [30]Incesu M., Gursoy O. The similarity invariants of Bézier curves and surfaces, Nat. Math. Symp., Ataturk University, Erzurum, Turkey, 2007.
30. [31]Samanci H.K., Celik S., Incesu M. The Bishop Frame of Bézier Curves, Life Sci. J., 12:6 175-180, 2015.
31. [32]Bulut V., Caliskan A. The Exchange Variations Between Bézier Directrix Curves of Two Developable Ruled Surfaces, J. Dynamical Sys. Geom. Theories, 13 103-114, 2015.
32. [33]Samanci H.K. Generalized dual-variable Bernstein polynomials, Konuralp J. Math., 5 56-67, 2016.
33. [34]Incesu M., Gursoy O. The similarity invariants of integral B-splines, International scientific conference Algebraic and geometric methods of analysis, May 31-June 5, Odesa, 2017.
34. [35]Erkan E., Yuce S. Serret-Frenet frame and curvatures of Bézier curves, Mathematics, 6 321-351, 2018.
35. [36]Samanci H.K. Some geometric properties of the spacelike Bézier curve with a timelike principal normal in Minkowski 3-space, Cumhuriyet Sci. J., 39 71-79, 2018.
36. [37]Samanci H.K., Kalkan O., Celik S. The timelike B´ezier spline in Minkowski 3-space, J. Sci. Arts, 19 357-374, 2019.
37. [38]Baydas S., Karakas B. Detecting a curve as a Bézier curve, J. Taibah Univ. Sci., 13 522-528, 2019.
38. [39]Marsh D. Applied geometry for computer graphics and CAD, Springer-Verlag London Berlin Heidelberg, London, 1999.