Planar curved creases represent an important family of creases and offer significant potential for application in interactive design tools. Once a folded state is constructed, the question arises: Does a folding motion exist, maintaining the same ruling layout, that connects the flat state to the folded state? This is referred to as a rigid-ruling folding motion. In this paper, we characterize combinations of planar curves that allow a rigid-ruling folding motion and illustrate our theoretical findings with examples.