Equivalent curves in ${\mathbb{E}}^n$

In this paper, we first define an equivalence relation for curves in ${\mathbb{E}}^n$. Based on this equivalence relation, we investigate the relationships between the Frenet frame and curvatures of equivalent curves. Next, we introduce the concept of linearly dependent curvatures in ${\mathbb{E}}^n$ and examine its implications for equivalent curves. Building on this concept and the proposed equivalence relation, we present a method to construct (1, 3)-Bertrand curves in ${\mathbb{E}}^4$. Additionally, we derive the relationships between the harmonic curvatures of equivalent curves and use these relationships to establish several properties of equivalent helical curves. These results enable systematic construction of curves with prescribed geometric properties.