In this work, we prove that the ratio of torsion and curvature of any dual rectifying curve is a non-constant linear function of its dual arc length parameter. Thereafter, a dual differential equation of third order is constructed for every dual curve. Then, several well-known characterizations of dual spherical, normal and rectifying curves are consequences of this differential equation. Finally, we prove a simple new characterization of dual spherical curves in terms of the Darboux vector.
Citation: Rashad Abdel-Baky, Mohamed Khalifa Saad. Some characterizations of dual curves in dual 3-space $ \mathbb{D}^{3} $[J]. AIMS Mathematics, 2021, 6(4): 3339-3351. doi: 10.3934/math.2021200
In this work, we prove that the ratio of torsion and curvature of any dual rectifying curve is a non-constant linear function of its dual arc length parameter. Thereafter, a dual differential equation of third order is constructed for every dual curve. Then, several well-known characterizations of dual spherical, normal and rectifying curves are consequences of this differential equation. Finally, we prove a simple new characterization of dual spherical curves in terms of the Darboux vector.
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Rashad Abdel-Baky, Mohamed Khalifa Saad. Some characterizations of dual curves in dual 3-space $ \mathbb{D}^{3} $[J]. AIMS Mathematics, 2021, 6(4): 3339-3351. doi: 10.3934/math.2021200
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