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Systems and methods for calculating a modification of a geometrical shape by applying an inverse modification function to an array representing the shape. An array representing the geometrical shape is defined on a multi-dimensional space. A modification function is used to modify the geometrical sh

Systems and methods for calculating a modification of a geometrical shape by applying an inverse modification function to an array representing the shape. An array representing the geometrical shape is defined on a multi-dimensional space. A modification function is used to modify the geometrical shape. A user or a programmed computer can select the modification function. The computer applies an inverse of the modification function to the array. The computer deduces a change in the geometrical shape from the modified array. An advantage of the system and method is the ability to compute shape changes in certain situations where a direct computation is cumbersome or otherwise inconvenient.

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1. A method of modifying a geometrical shape, comprising the steps of:defining in a multi-dimensional space an array of values representing a geometrical shape; selecting a modification function that represents a desired modification to be applied to the geometrical shape; applying an inverse functi

1. A method of modifying a geometrical shape, comprising the steps of:defining in a multi-dimensional space an array of values representing a geometrical shape; selecting a modification function that represents a desired modification to be applied to the geometrical shape; applying an inverse function of the modification function to the array of values to produce a modified array; deducing from the modified array a modification of the geometrical shape that would result from a direct application of the modification function to the array; and applying the resulting modification of the geometrical shape by manipulation of a virtual tool, wherein the geometrical shape is displaced with respect to the virtual tool. 2. The method of claim 1, further comprising displaying to a user the modification of the geometrical shape that would result from a direct application of the modification function to the array.3. The method of claim 1, wherein the step of deducing from the modified array a modification further comprises the steps of:retrieving an array value from the modified array; and applying the array value from the modified array at a location in the array representing the geometrical shape. 4. The method of claim 1, wherein the modification function comprises a soft-edged deformation.5. The method of claim 1, wherein the modification function is applied from a selected one of a first side and a second side of the geometrical shape.6. The method of claim 1, wherein the geometrical shape is displaced away from the virtual tool.7. The method of claim 1, wherein the geometrical shape is displaced toward the virtual tool.8. The method of claim 7, wherein the modification of the geometrical shape is substantially a convex hump.9. The method of claim 7, wherein the modification of the geometrical shape is substantially a concave spiked protuberance.10. The method of claim 1, wherein the modification function comprises a force field consistent with a tool of arbitrary shape.11. The method of claim 1, wherein the modification function comprises translational displacement.12. The method of claim 1, wherein the modification function comprises rotational displacement.13. The method of claim 1, wherein the modification function comprises a selected one of a displacement function, a smoothing function, a warping function, a volumetric interference, an areal interference, a result of a simulation, a data re-fitting, and a force.14. The method of claim 1, wherein the modification function is represented as a non-linear mathematical function.15. The method of claim 1, further comprising the steps of:selecting a second modification function; applying an inverse function of the second modification function to the modified array to produce a twice-modified array; and deducing from the twice-modified array a modification of the geometrical shape that would result from an application of the modification function to the array followed by an application of the second modification function to the once-modified array. 16. The method of claim 1, wherein the modification comprises applying a constraint so as to control a magnitude of a change of the geometrical shape.17. The method of claim 16, wherein applying the constraint prevents at least one point of the geometrical shape from moving in response to the application of the inverse function of the modification function.18. The method of claim 1, wherein the modification comprises applying a surface texture to the geometrical shape.19. A system for modifying a geometrical shape, comprising:a generation module that defines in a multi-dimensional space an array of values representing a geometrical shape; a selection module that provides a modification function representing a desired modification to be applied to the geometrical shape; a transformation module that applies an inverse function of the modification function to the array of values to produce a modified array; a calculation module that deduces from the modified array a modification of the geometrical shape that would result from a direct application of the modification function to the array; and a modification module that applies the resulting modification of the geometrical shape by manipulation of a virtual tool, wherein the geometrical shape is displaced with respect to the virtual tool. 20. The system of claim 19, further comprising a display module that displays to a user the modification of the geometrical shape that would result from a direct application of the modification function to the array.21. The system of claim 19, wherein the calculation module further comprises:a module that retrieves an array value from the modified array; and a module that applies the array value from the modified array at a location in the array representing the geometrical shape. 22. The system of claim 19, wherein the modification function comprises a soft-edged deformation.23. The system of claim 19, wherein the modification function is applied from a selected one of a first side and a second side of the geometrical shape.24. The system of claim 19, wherein manipulation of a virtual tool displaces the geometrical shape away from the virtual tool.25. The system of claim 19, wherein manipulation of a virtual tool displaces the geometrical shape toward the virtual tool.26. The system of claim 25, wherein the modification of the geometrical shape is substantially a convex hump.27. The system of claim 25, wherein the modification of the geometrical shape is substantially a concave spiked protuberance.28. The system of claim 19, wherein the modification function comprises a force field consistent with a tool of arbitrary shape.29. The system of claim 19, wherein the modification function comprises translational displacement.30. The system of claim 19, wherein the modification function comprises rotational displacement.31. The system of claim 19, wherein the modification function comprises a selected one of a displacement function, a smoothing function, a warping function, a volumetric interference, an areal interference, a result of a simulation, a data re-fitting, and a force.32. The system of claim 19, wherein the modification function is represented as a non-linear mathematical function.33. The system of claim 19, further comprising:a selection module that selects a second modification function; a transformation module that applies an inverse function of the second modification function to the modified array to produce a twice-modified array; and a calculation module that deduces from the twice-modified array a modification of the geometrical shape that would result from an application of the modification function to the array followed by an application of the second modification function to the once-modified array. 34. The system of claim 19, wherein the modification comprises applying a constraint so as to control a magnitude of a change of the geometrical shape.35. The system of claim 34, wherein the module that applies the constraint prevents at least one point of the geometrical shape from moving in response to the application of the inverse function of the modification function.36. The system of claim 19, wherein the modification comprises applying a surface texture to the geometrical shape.