TY - JOUR T1 - New vector fields and planes of framed curves in Euclidean 4-space AU - Yıldız, Önder Gökmen AU - Balkan, Fatma PY - 2025 DA - November Y2 - 2025 JF - Düzce Mathematical Research PB - Duzce University WT - DergiPark SN - 3108-5741 SP - 34 EP - 40 VL - 1 IS - 1 LA - en AB - In this study, we define new Darboux vectors for curves with singular points in Euclidean 4-space. By using these vectors, we construct new planes and determine curves lying in these planes. Subsequently, we give characterizations and corollaries related to these curves. KW - Darboux vector KW - framed curves KW - singular points CR - [1] S. Izumiya, N. Takeuchi (2004), New special curves and developable surfaces, Turk. J. Math., 28(2), 153–163. CR - [2] Ç. Çamcı, K. İlarslan, L. Kula, H.H. Hacısalihoğlu (2009), Harmonic curvatures and generalized helices in En, Chaos Solitons Fractals, 40(5), 2590–2596. https://doi.org/10.1016/j.chaos.2007.11.001 CR - [3] M. Düld¨ül (2020), Vector fields and planes in E4 which play the role of Darboux vector, Turk. J. Math., 44(1), 274–283. https://doi.org/10.3906/mat-1908-9 CR - [4] S. Honda, M. Takahashi (2016), Framed curves in the Euclidean space, Adv. Geom., 16(3), 265–276. https://doi.org/10.1515/advgeom-2015-0035 CR - [5] S. Honda, M. Takahashi (2016), Evolutes of framed immersions in the Euclidean space, Hokkaido Univ. Preprint Series in Math., 1095, 1–24. https://doi.org/10.14943/84239 CR - [6] T. Fukunaga, M. Takahashi (2013), Existence and uniqueness for Legendre curves, J. Geom., 104(2), 297–307. https://doi.org/10.1007/s00022-013-0162-6 CR - [7] K. Eren, Ö.G. Yıldız, M. Akyiğit (2022), Tubular surfaces associated with framed base curves in Euclidean 3-space, Math. Methods Appl. Sci., 45(18), 12110–12118. https://doi.org/10.1002/mma.7590 CR - [8] M. Ateş, M. Akyiğit (2023), Framed general helix and framed z3-slant helix in R4, An. Ştiin. Univ. Ovidius Constanta Ser. Mat., 31(3). https://doi.org/10.2478/auom-2023-0029 CR - [9] B. Doğan Yazıcı, O.Z. Okuyucu, M. Tosun (2023), On special singular curve couples of framed curves in 3D Lie groups, Commun. Fac. Sci. Univ. Ankara Ser. A1 Math. Stat., 72(3), 710–720. https://doi.org/10.31801/cfsuasmas.1197154 CR - [10] O.Z. Okuyucu, M. Canbirdi (2021), Framed slant helices in Euclidean 3-space, Adv. Differ. Equ., 2021, 1–14. https://doi.org/10.1186/s13662-021-03664-7 CR - [11] Ö.G. Yıldız, M. Akyiğit, M. Tosun (2021), On the trajectory ruled surfaces of framed base curves in the Euclidean space, Math. Methods Appl. Sci., 44(9), 7463–7470. https://doi.org/10.1002/mma.6267 CR - [12] B.D. Yazıcı, S.Ö. Karakuş, M. Tosun (2022), On framed Tzitzeica curves in Euclidean space, Facta Univ. Ser. Math. Inform., 37(2), 307–319. https://doi.org/10.22190/FUMI211025021D CR - [13] B.D. Yazıcı, S.Ö. Karakuş, M. Tosun (2022), On the classification of framed rectifying curves in Euclidean space, Math. Methods Appl. Sci., 45(18), 12089–12098. https://doi.org/10.1002/mma.7561 CR - [14] B.D. Yazıcı, S.Ö. Karakuş, M. Tosun (2021), Framed normal curves in Euclidean space, Tbilisi Math. J., 14(3), 27–37. https://doi.org/10.1002/mma.7561 CR - [15] S. Honda, Flat surfaces associated with framed base curves, Ph.D. Thesis, Hokkaido University, Japan, 2018. https://doi.org/10.14943/doctoral.k13123 CR - [16] M.Z. Williams, F.M. Stein (1964), A triple product of vectors in four-space, Math. Mag., 37(4), 230–235. https://doi.org/10.1080/0025570X.1964.11975525 CR - [17] M.N. Şavlı, 4-boyutlu Öklid uzaynda çatılandırılmış oskültatör ve rektifiyan eğriler, Masters thesis, Bilecik Şeyh Edebali University, Türkiye, 2022. CR - [18] B.Y. Chen (2003), When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly, 110(2), 147–152. https://doi.org/10.2307/3647775 UR - https://dergipark.org.tr/en/pub/dmr/issue//1737129 L1 - https://dergipark.org.tr/en/download/article-file/5033148 ER -