Achieving curved crease surfaces is a current challenge for designers, the field still underexplored. The curve on which the bending is performed can present extremely complicated shapes that must be accurately generated through various technological processes. For this aim, a planar mechanism consisting of an ellipsograph, a conchoidograph and an RRR dyad—whose inner joint generates the targeted connecting rod curve—was proposed in this paper. Parametrizing the lengths of the elements of the RRR dyad, the correlation coefficient of the rotations of the leading elements and the directions of these movements result in different connecting rod curves, which can be used to obtain the folded surfaces; the optimization, according to various criteria, will be explored for the automatic generation of some design solutions. With the help of the Altair program, for some given geometrical data of the mechanism, both the drawing of the curve on plates of different initial shapes and the simulation of their folding by applying compression forces on the bounding surfaces were carried out. By cutting the deformed shape, folded surface units used in tessellations are obtained.