TY - JOUR T1 - Why Flc-Frame is Better than Frenet Frame on Polynomial Space Curves? AU - Dede, Mustafa PY - 2022 DA - December Y2 - 2022 JF - Mathematical Sciences and Applications E-Notes JO - Math. Sci. Appl. E-Notes PB - Murat TOSUN WT - DergiPark SN - 2147-6268 SP - 190 EP - 198 VL - 10 IS - 4 LA - en AB - It is well known that the binormal and normal vectors of Frenet frame rotate around the tangent vector. That is why the Frenet frame is not suitable for some applications such as tube surfaces. However, there is not enough information about why the vectors of the Frenet frame rotate around the tangent vector. In this paper we will deal with this problem. Moreover we show the advantages of Flc-frame over the Frenet frame. KW - Frenet frame KW - space curve KW - adapted frame CR - [1] Bishop, R. L.: There is more than one way to frame a curve. Amer. Math. Monthly 82, 246–251 (1975). CR - [2] Bloomenthal, J.: Calculation of reference frames along a space curve. Graphics gems, Academic Press Profes- sional, Inc., San Diego, CA (1990). CR - [3] Guggenheimer, H.: Computing frames along a trajectory. Comput. Aided Geom. Des. 6, 77–78 (1989). CR - [4] Wang, W. Juttler, B. Zheng D. and Liu Y.: Computation of rotation minimizing frame. ACM Trans. Graph. 27(1) (2008). CR - [5] DoCarmoM.P.:DifferentialGeometryofCurvesandSurfaces,PrenticeHall,EnglewoodCliffs,NJ(1976). CR - [6] Mäurer, C. and Jüttler, B.: Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics. Journal for Geometry and Graphics, 3(2), 141-159 (1999). CR - [7] Klok, F.: Two moving coordinate frames for sweeping along a 3D trajectory. Comput. Aided Geom. Des. 3, 217–229 (1986). CR - [8] Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition, CRC Press, Boca Raton (1998). CR - [9] Jüttler, B. and Mäurer, C.: Cubic Pythagorean Hodograph Spline Curves and Applications to Sweep Surface Modeling. Comput. Aided Design. 31, 73-83 (1999). CR - [10] Ravani, R. Meghdari A. and Ravani, B.: Rational Frenet-Serret curves and rotation minimizing frames in spatial motion design. IEEE international conference on Intelligent engineering systems, INES 186-192 (2004). CR - [11] Dede, M. Ekici, C. and Görgülü, A.: Directional q-frame along a space curve. IJARCSSE. 5(12), 775-780 (2015). CR - [12] Ganovelli, F., Corsini, M., Pattanaik, S., Di Benedetto, M.: Introduction to Computer Graphics: A Practical Learning Approach. CRC Press, Boca Raton (2014). CR - [13] Farouki, R. T.: Pythagorean-hodograph curves: Algebra and Geometry. Springer (2008). CR - [14] Hanson, A. J. , Quaternion Frenet frames: making optimal tubes and ribbons from curves, Tech. Rep. 407, Indiana University Computer Science Department, (1994). CR - [15] Dede, M.: A New Representation of Tubular Surfaces. Houston Journal of Mathematics. 45(3), 707-720 (2019). CR - [16] Shen, L.Y., Yuan, C.M., Gao, X.S.: Certified approximation of parametric space curves with cubic B-spline curves. Computer Aided Geometric Design. 29(8), 648-663 (2012). CR - [17] Dede, M.: On polynomial space curves, preprint. CR - [18] Bukcu, B. and Karacan, M. K.: On The Modified Orthogonal Frame with Curvature and Torsion in 3-Space. Mathe- matical Sciences and Applications E-Notes. 4(1), 184-188 (2016). CR - [19] Lone,M.S.,EsH.,Karacan,M.K.,Bükcü,B.:MannheimcurveswithmodifiedorthogonalframeinEuclidean3-space, Turkish Journal of Mathematics. Turk J Math. 43, 648-663 (2019). CR - [20] Dede, M. Ekici, C., Tozak, H.: Directional Tubular Surfaces. International Journal of Algebra, 9, 527-535 (2015). CR - [21] Korpinar, T., Bas, S.: Directional Inextensible Flows of Curves by Quasi Frame, J. Adv. Phys. 7, 427-429 (2018). CR - [22] S ̧enyurt, S., Ayvacı, H., Hilal Ayvacı, Canli, D.,: Some characterizations of spherical indicatrix curves generated by Flc frame, Turk. J. Math. Comput. Sci. 13, 379–387 (2021). CR - [23] S ̧enyurt, S., Sardag ̆, H., Çakır, O.: On vectorial moment of the Darboux vector, Konuralp J. Math. 8, 3 144–151. (2020). CR - [24] Körpınar,T.,Sazak,A.,Körpınar,Z.:Opticaleffectsofsomemotionequationsonquasi-framewithcompatibleHasimoto map, Optik 247, 167914 (2021). CR - [25] Körpınar, T., Körpınar, Z., Asil, V.: Optical effects of some motion equations on quasi-frame with compatible Hasimoto map, Optik 251, 168291 (2022). UR - https://dergipark.org.tr/en/pub/mathenot/issue//641495 L1 - https://dergipark.org.tr/en/download/article-file/846045 ER -