Some integral curves according to quasi-frame in Euclidean 3-space

Abstract

This study explores integral curves associated with the quasi-frame in three-dimensional Euclidean space. We focus specifically on the quasi-normal and quasi-binormal vectors. We derive the Frenet apparatus for these integral curves based on the quasi-frame elements. Our analysis reveals significant relationships between the integral curves and the original curve’s Frenet frame. We present explicit expressions for the Frenet–Serret apparatus of both quasi-normal and quasi-binormal curves. Moreover, we identify conditions under which these integral curves qualify as general helices or Salkowski curves. The study examines geometric relationships, including involute-evolute pairs and Bertrand pairs. Additionally, we analyze conditions that prevent the formation of Mannheim curve pairs.