Methods and system for computing integrals over geometric domains using moment-base representations for interoperability are disclosed. A first computing device may receive data for a geometric representation of an object, the geometric representation specifying a geometric domain corresponding to a shape and a contained spatial region of the object. The first computing device may computationally integrate a predetermined set of basis functions over the geometric domain to derive a moment-vector for the object, the moment-vector encapsulating an analytic formulation of the geometric domain that is independent of the geometric representation. The first computing device may then computationally generate quadrature rules for integrating an arbitrary function by applying moment-fitting to the moment-vector. The quadrature rules may be provided to a second computing device. The second computing device may computationally integrate the arbitrary function over the geometric domain by applying the quadrature rules, independently of the geometric representation of the object.

A method for automatic creation of cutting paths 5 in the interior space 2 of a three-dimensional shaped product 1 by the following steps, in a CAD/CAM system in which programs are designed for lamination, sintering and cutting that are necessary for creating a three-dimensional shaped product 1: defining an imaginary horizontal plane 3, creating cutting paths 5 in the interior wall section from the lower end successively toward the upper side, creating cutting paths 5 in the interior wall section toward the upper side to the upper end, and joining the cutting paths 5 of step 2 and the cutting paths 5 of step 3.

A method for machining a workpiece using a programmable, numerically controlled machining system by calculating or retrieving a compensated toolpath based on comparing contact position from monitoring a vibration signal from a vibration sensor during probing of workpiece with rotating tool during relative motion therebetween. Contact position is compared to position from predetermined toolpath and wherein the predetermined toolpath extends between initial machining point and end machining point. Machining the workpiece is done along compensated toolpath. The method may be done for repeated passes of machining. The compensated toolpath may include an angle offset to a machining path coordinate system of the predetermined toolpath. Workpiece may be mounted in a multi-axis manipulator of machining system for the probing and machining Multi-axis manipulator may be computer controlled and may be part of a robot.