When reviewing the development of the generalized theory of capillarity, we find that the current theories for contact lines require significant modifications. The choice of parameters (independent variables) for a line is often insufficient for formulating the thermodynamic fundamental equation. Furthermore, there are differential geometric constraints on the geometric parameters but not included in the analyses. As a result, the independent variables in the fundamental equation of the current theories are in fact dependent. To describe the geometry of contact lines properly, we present a new set of parameters from the differential geometry viewpoint and subsequently give the fundamental equation for the thermodynamic system of contact lines. We will further do variational problems to obtain the descriptive formalism of equilibrium conditions of contact line systems.
