This paper delves into the recent advancements of log-aesthetic curves (LAC), a family of curves for industrial design. Recently it has been redefined in the context of similarity geometry, a branch of Klein geometry. Building upon the work of Miura and Gobithaasan [1], the paper explores LAC properties, delves into the connection with similarity geometry (as detailed in Inoguchi et al. [2-5]), and examines extensions of LACs, including the σ-curve [6] and τ-curve [7].