In 1960, E. H. Brown defined a set of characteristic curves (also known as ideal curves) of pure fluids, along which some thermodynamic properties match those of an ideal gas. These curves are used for testing the extrapolation behaviour of equations of state. This work is revisited, and an elegant representation of the first-order characteristic curves as level curves of a master function is proposed. It is shown that Brown’s postulate—that these curves are unique and dome-shaped in a double-logarithmic p ,  T representation—may fail for fluids exhibiting a density anomaly. A careful study of the Amagat curve (Joule inversion curve) generated from the IAPWS-95 reference equation of state for water reveals the existence of an additional branch.