Methods and apparatus for defining a low-curvature tool path

A method for defining a path that a cutting tool is capable of traveling along to remove a region of material having an outer boundary, the method comprising:

• determining an offset boundary that the outer boundary extends around;
• determining a plurality of nested contours that are internal to the outer boundary, wherein the determining the plurality of nested contours comprises:
• determining a mathematical function that at least approximates the solution of an elliptic partial differential equation boundary value problem defined on the region enclosed by the offset boundary, and
• determining from the mathematical function the plurality of nested contours;
• and
• spiraling between the contours to at least partially form the path.

A method for defining a path that a cutting tool is capable of traveling along, the method comprising:

• determining a boundary that extends around and encloses a region;
• determining a mathematical function that at least approximates a solution of an elliptic partial differential equation boundary value problem defined on the region enclosed by the boundary;
• determining from the mathematical function a plurality of nested contours;
• and
• moving between the contours to at least partially form the path.

A machine capable of moving a cutting tool to remove a region of material having an outer boundary, the machine comprising:

• a tool holder capable of holding the cutting tool;
• and
• a controller operative for moving the tool holder along a path while the tool holder holds the cutting tool, wherein the controller comprises a computer-readable medium on which is stored a program module for defining the path, the program module comprising instructions which, when executed by the controller, perform operations comprising:
• determining a plurality of nested contours that are internal to the outer boundary;
• and
• spiraling between the contours to at least partially form the path, wherein the spiraling comprises forming a first spiral that extends from proximate a first one of the contours to proximate a second one of the contours, and wherein the forming comprises calculating curvature of the first spiral as a function of both of the first and second contours.

A machine capable of moving a cutting tool, the machine comprising:

• a tool holder capable of holding the cutting tool;
• and
• a controller operative for moving the tool holder along a path while the tool holder holds the cutting tool, wherein the controller comprises a computer-readable medium on which is stored a program module for defining the path, the program module comprising instructions which, when executed by the controller, perform operations comprising:
• determining a boundary that extends around and encloses a region,
• determining a mathematical function that at least approximates a solution of an elliptic partial differential equation boundary value problem defined on the region enclosed by the boundary, and
• determining from the mathematical function a plurality of nested contours.

A computer-readable medium on which is stored a program module for defining a path that a cutting tool is capable of traveling along to remove a region of material having an outer boundary, the program module comprising instructions which, when executed by a computer, perform operations comprising:

• determining an offset boundary that the outer boundary extends around;
• determining a plurality of nested contours that are internal to the outer boundary, wherein the determining the plurality of nested contours comprises:
• determining a mathematical function that at least approximates the solution of an elliptic partial differential equation boundary value problem defined on the region enclosed by the offset boundary, and
• determining from the mathematical function the plurality of nested contours;
• and
• spiraling between the contours to at least partially form the path.

A computer-readable medium on which is stored a program module for defining a path that a cutting tool is capable of traveling along, the program module comprising instructions which, when executed by a computer, perform operations comprising:

• determining a boundary that extends around and encloses a region;
• determining a mathematical function that at least approximates a solution of an elliptic partial differential equation boundary value problem defined on the region enclosed by the boundary;
• determining from the mathematical function a plurality of nested contours;
• and
• moving between the contours to at least partially form the path.