The NURBS book
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A new approach to model the actual machining result as a NURBS surface is presented, which explicitly expresses the geometry and topology of the final product and increases the clarity in the mathematical representation of quasistatic machining errors. Most of the available models that estimate interaction of quasistatic machining errors present the actual position of individual machined points and are unable to explicitly describe the resulting machined surface. During the machining process, the desired geometry is mapped from the ideal computer-aided design (CAD) vector space into the machine tool's vector subspaces. Using a Jacobian of the deformed geometry, it is shown that for a variety of three-axis machine tool configurations, a linear operator can be found to express this transformation. Classified error operators for all different configurations of three-axis machine tools are derived and the applicability of the developed method is illustrated by simulating the machining process using case studies. The developed model can be utilised in virtual machining and simulation of the machining process, modification of the design within a design for a manufacturing platform, and also in on-line error compensation during the machining process.