Before revision:  In this paper, we show that the case of four mutually non-congruent curved foldings with the same crease and crease pattern occurs if and only if  \Gamma and C do not admit any symmetries.
After revision:  In this paper, we show that the case of four mutually non-congruent curved foldings with the same crease and crease pattern . As a consequence, if C is a non-closed simple arc, then N=4 if and only if \Gamma and C do not admit any symmetries.