When the charge on a planar conductor is a function of its curvature

While there is no general relationship between the electric charge density on a conducting surface and its curvature, the two quantities can be functionally related in special circumstances. This paper presents a complete classification of two-dimensional conductors for which charge density is a function of boundary curvature. Whenever the curvature function is non-injective, the conductor must transform under one of the planar symmetry groups. In particular, for the charge density on a closed conductor with smooth boundary to be a function of its curvature, the conductor must possess dihedral symmetry with a mirror line running through each curvature extremum. Several examples are presented along with explicit charge-curvature functions. Both increasing and decreasing functions were found.