Some Special Legendre Mates of Spherical Legendre Curves

Abstract

In this study, we consider some special mates of spherical Legendre curves by using Legendre frame along spherical front or frontal on Euclidean unit sphere. In this sense, we define orthogonal-type and parallel-type spherical Legendre mates. After, we get some characterizations between Legendre curvatures of orthogonal-type Legendre mates. In particularly, when spherical front is a regular curve, we give relationships between Legendre and Frenet frame or Legendre and Sabban frame. Moreover, we obtain that the evolute and the involute of spherical front correspond to a second and third orthogonal-type Legendre mate of spherical front, respectively. After, we give examples for first, second or third orthogonal-type spherical Legendre mates of some special spherical fronts or frontals. Finally, we show that there is no parallel-type Legendre mates of spherical frontal.