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## Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications

#### Fırat YERLİKAYA [1] , İsmail AYDEMİR [2]

We analyze integrability for the derivative formulas of the rotation minimizing frame in the Euclidean 3-space from a viewpoint of rotations around axes of the natural coordinate system. We give a theorem that presents only one component of the indirect solution of the rotation minimizing formulas. Using this theorem, we find a lemma which states the necessary condition for the indirect solution to be a steady solution. As an application of the lemma, the natural representation of the position vector field of a smooth curve whose the rotation minimizing vector field (or the Darboux vector field) makes a constant angle with a fixed straight line in space is obtained. Also, we realize that general helices using the position vector field consist of slant helices and Darboux helices in the sense of Bishop.
Bishop slant helix, Darboux helix, rotation minimizing frame, Euclidean 3-space
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Primary Language en Mathematics Research Article Orcid: 0000-0003-2360-1522Author: Fırat YERLİKAYA (Primary Author)Institution: ONDOKUZ MAYIS UNIVERSITYCountry: Turkey Orcid: 0000-0002-0238-2079Author: İsmail AYDEMİR Institution: ONDOKUZ MAYIS UNIVERSITYCountry: Turkey Publication Date : January 30, 2020
 Bibtex @research article { iejg621588, journal = {International Electronic Journal of Geometry}, issn = {}, eissn = {1307-5624}, address = {}, publisher = {Kazım İLARSLAN}, year = {2020}, volume = {13}, pages = {116 - 128}, doi = {10.36890/iejg.621588}, title = {Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications}, key = {cite}, author = {YERLİKAYA, Fırat and AYDEMİR, İsmail} } APA YERLİKAYA, F , AYDEMİR, İ . (2020). Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications. International Electronic Journal of Geometry , 13 (1) , 116-128 . DOI: 10.36890/iejg.621588 MLA YERLİKAYA, F , AYDEMİR, İ . "Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications". International Electronic Journal of Geometry 13 (2020 ): 116-128 Chicago YERLİKAYA, F , AYDEMİR, İ . "Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications". International Electronic Journal of Geometry 13 (2020 ): 116-128 RIS TY - JOUR T1 - Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications AU - Fırat YERLİKAYA , İsmail AYDEMİR Y1 - 2020 PY - 2020 N1 - doi: 10.36890/iejg.621588 DO - 10.36890/iejg.621588 T2 - International Electronic Journal of Geometry JF - Journal JO - JOR SP - 116 EP - 128 VL - 13 IS - 1 SN - -1307-5624 M3 - doi: 10.36890/iejg.621588 UR - https://doi.org/10.36890/iejg.621588 Y2 - 2019 ER - EndNote %0 International Electronic Journal of Geometry Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications %A Fırat YERLİKAYA , İsmail AYDEMİR %T Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications %D 2020 %J International Electronic Journal of Geometry %P -1307-5624 %V 13 %N 1 %R doi: 10.36890/iejg.621588 %U 10.36890/iejg.621588 ISNAD YERLİKAYA, Fırat , AYDEMİR, İsmail . "Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications". International Electronic Journal of Geometry 13 / 1 (January 2020): 116-128 . https://doi.org/10.36890/iejg.621588 AMA YERLİKAYA F , AYDEMİR İ . Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications. Int. Electron. J. Geom.. 2020; 13(1): 116-128. Vancouver YERLİKAYA F , AYDEMİR İ . Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications. International Electronic Journal of Geometry. 2020; 13(1): 128-116.

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