Curves orthogonal to a vector field in Euclidean spaces

Luiz C. B. da Silva, Gilson S. Ferreira Jr

Research output: Contribution to journalArticlepeer-review


A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curves that are also slant helices in three-dimensional space as geodesics of circular cones. In addition, we consider curves that lie on a moving hyperplane normal to (i) one of the normal vector fields of the Frenet frame and to (ii) a rotation minimizing vector field along the curve. The former class is characterized in terms of the constancy of a certain vector field normal to the curve, while the latter contains spherical and plane curves. Finally, we establish a formal mapping between rectifying curves in an -dimensional space and spherical curves in an -dimensional space.
Original languageEnglish
Pages (from-to)1485-1500
Number of pages16
JournalJournal of the Korean Mathematical Society
Issue number6
Publication statusPublished - 1 Nov 2021


  • Rectifying curve
  • Geodesic
  • Cone
  • Spherical curve
  • Plane curve
  • Slant helix


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