Curvature equivalence for Legendre curves in the unit tangent bundle over Euclidean plane

The Legendre curve in the unit tangent bundle over Euclidean plane is a plane curve with a moving frame. We have the (Legendre) curvature of the Legendre curve, and the existence and uniqueness theorems for the curvature are valid. In this paper, we introduce an equivalence relation for Legendre curves called the curvature equivalence. We investigate properties of the curvature equivalence and give local and global classifications of Legendre curves under the curvature equivalence.