This article presents a comparative study of Kiepert’s trefoil and its related curves, combining a variety of tools from differential and algebraic geometry, integrable systems, elastica theory, and special functions. While this curve was classically known and well studied in the literature, some related open problems were recently solved, and the goal of this paper is to present and characterize the general solution of the equation that governs this trefoil’s family of curves by involving elliptic functions and elastica theory in the mechanics.
