Die and backer, constantly facing each other, travel in cooperating paths which bring them closer to each other with sheet material sandwiched between, all traveling at the same speed in the same direction, and then together to effect a contour cut. The die may move in a circular path or a quasi elliptical path and the backer may move in a similar path to the die or may move in a linear path. The two sides of the backer may be somewhat relatively dephased to produce a rocking component of motion and a rocking cut. Specific preferred structure includes shafts in sets of three and transmission of motion from below the sheet material to above it by pairs of lever arms attached to rollers in roller guides.