Epicycloids generating Hamiltonian minimal surfaces in the complex quadric

We show that Hopf tubes on Lancret curves shaped over an epicycloid are Hamiltonian minimal surfaces in the complex quadric. Moreover they are the only Hopf tubes that are Hamiltonian minimal there. This allows one to connect two apparently unrelated topics, such as Hamiltonian minimal surfaces and curves with constant precession, and more generally slant helices. Furthermore, Hamiltonian minimal Hopf tubes encode the phases of particles described according to the gyroscopic force theory.