Mesh Surface Contour-Parallel Tool Path Based On Least Squares Conformal Map

Currently, mesh surfaces have received great attention due to its extensive application for data exchange and geometric computation. However, for mesh surface machining, there are few tool path planning strategies in addition to iso-planar method. Thus, in this thesis, a contour-parallel tool path method is presented for three-axis NC machining of complex mesh surfaces with holes or islands based on least squares conformal mapping.First, a least squares conformal map method is introduced, which is widely used in the texture mapping field. Based on a least-squares approximation of the Cauchy-Riemann equations, an energy function is defined minimizing angle deformations. Then by solving a system of linear equations, a mapping relationship is constructed between3D mesh surface and planar parameter domain, thus reducing the task of generating contour-parallel tool paths from3D mesh surface to the planar region. As LSCM has the property of the conformality, original mesh topology structure is well preserved in the planar parametric domain.Secondly, contour-parallel tool path is generated in the plane mesh. An efficient method named bounding box equidistance is presented for complex2D contour offsetting. One to one correspondence relation is constructed between offsetting segment and bounding box by dividing the box. This method can efficiently calculate intersection between the mass lines and eliminate unstable situation in the process of a large number of short line segments offsetting. After local interference and global interference are simultaneously removed by invoking the effective loops extracting method, interference-free contour-parallel tool paths are finally generated.Moreover, a space partitioning strategy is used to determine which triangle of parameter domain planar point is located in. To avoid calculating inverse matrix, a area coordinate method is presented to inversely map the planar contour-parallel point onto3D mesh surface. Cutter contact point is converted into cutter location point according to cutting tool’s parameters. Finally, complete contour-parallel tool paths are generated for complex mesh surface. Finally, all of the algorithms are coded and implemented in C++language. The contour-parallel tool paths for machining a joystick are simulated in the UG. The result of simulation verifies the feasibility of the proposed method.