Generating a plurality of curved transitions connecting planar parallel alternating paths for forming a workpiece

A system comprises:

• a three-dimensional computer model for a workpiece comprising a desired shape of the workpiece;
• a tool path generator in a computer system, the tool path generator configured to:
• divide the desired shape of the workpiece into a plurality of layers stacked on top of one another, define a plurality of alternating paths that each circumferentially traverses a fraction of a respective layer of the plurality of layers including a first path that circumferentially traverses a fraction of a first layer of the plurality of layers, a second path that circumferentially traverses a fraction of a second layer of the plurality of layers, and a third path that circumferentially traverses a fraction of a third layer of the plurality of layers,
• calculate a plurality of curved transitions, wherein each curved transition of the plurality of curved transitions joins two respective alternating paths of the plurality of alternating paths including calculating a curved transition that joins the first path to the second path and calculating a second curved transition that joins the second path to the third path, wherein the curved transition and the second curved transition are oriented in opposite directions, and
• generating a tool path containing the plurality of alternating paths and the plurality of curved transitions;
• a controller in the computer system, the controller in communication with the tool path generator, wherein the controller is configured to receive the tool path from the tool path generator, wherein the controller is configured to drive a forming tool along the plurality of alternating paths and the plurality of curved transitions to shape a material into the workpiece wherein driving the forming tool includes driving the forming tool along the tool path containing the first path, the curved transition, and the second path to shape a material into the workpiece, wherein driving the forming tool along the first path comprises moving the forming tool in a first direction, wherein driving the forming tool along the second path comprises moving the forming tool in a second direction, and wherein the first direction is opposite the second direction, and driving the forming tool further including driving the forming tool along the third path and the second curved transition to shape the material into the workpiece, wherein the curved transition and the second curved transition do not intersect, wherein the forming tool contacts a same portion of the material while driving the forming tool along the curved transition and while driving the forming tool along the second curved transition;
• and
• the forming tool in communication with the controller.