A digitally controlled machine for bending, forming, or curving raw parts, such as rods, wires or profiled sections, comprises a conventional main frame (1) and a rotating bending head capable of bending articles in different planes. The rod (4) is horizontally displaced along a transport axis (5) by a feed station, located upstream, and is brought through a bending shank (7) borne by the main frame (1). The rod is then brought through a lathe (10) of the bending head. The lathe is formed by a bending roll (8) and a folding pin (9), and is capable of rotating in a horizontal plane about an axis of rotation aligned with the axis of the bending roll (8). The bending shank (7) is movably mounted in translation in a transport axis (5) and is continuously displaced between one or more projecting bending positions wherein the shank is proximate to the roll (8) and one or more retracted positions, facing upstream, which completely clear the bending zone between the lathe and the main frame.