  We present a class of thermodynamic systems with constant thermodynamic
curvature which, within the context of geometric approaches of thermodynamics,
can be interpreted as constant thermodynamic interaction among their
components. In particular, for systems constrained by the vanishing of the
Hessian curvature we write down the systems of partial differential equations.
In such a case it is possible to find a subset of solutions lying on a
circumference in an abstract space constructed from the first derivatives of
the isothermal coordinates. We conjecture that solutions on the characteristic
circumference are of physical relevance, separating them from those of pure
mathematical interest. We present the case of a one-parameter family of
fundamental relations that -- when lying in the circumference -- describe a
polytropic fluid.
