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Constant cusp is a common strategy for generating tool paths in many NC machining applications. Cusps need to be regulated to ensure high precision without wasting machining efforts. Constant cusp strategies frequently operate on NURBS surfaces or triangular meshes and, thus, have to deal with the issues of patch-boundary oscillations or long, stretched triangles. To avoid these issues, one can operate in a pre computed configuration space (c-space). The c-space is given in form of a regular quadrilateral heightfield mesh, which may be adaptively subdivided, where the slope is large. This simple data structure is memory efficient and is widely used in CAD/CAM frameworks. In this paper we introduce an algorithm for creating a constant cusp tool path with the help of a given c-space. The constant cusp algorithm iteratively produces curves in the c-space by fitting a tube around the current curve and intersecting the tube with the c-space mesh to detect the subsequent curve. As tool paths are handed to the machine controller in form of point sequences, it suffices to operate on piecewise linear curves. The tube becomes a concatenation of cylinders, which we derive using geometric considerations. In each iteration of the constant cusp algorithm, intersection points of the cylinders with the not yet traversed part of the mesh are detected and checked for their validity. The validity check can efficiently remove global or local self-intersections of the new curve by just deleting the respective points. In a final step, the detected intersection points are connected to form constant cusp tool paths. Dealing with piecewise linear curves, we achieve low computation times for real-world data sets

An automated polishing process of free-form surfaces requires a tool path that covers the entire surface equally and forms an overlapping pattern without visible artifacts. The recently presented double-spiral tool paths assure a coverage of the entire surface with a continuous, non-overlapping path and low variation in distance between adjacent traces of the path. We build upon this approach by constructing cycloids of flexible radii that fill the space between adjacent traces. The use of cycloids mimics the cyclic movement when polishing by hand. The approach operates in a precomputed configuration space (c-space) given in form of an adaptive quadrilateral heightfield mesh. Operating in c-space avoids having to deal with the issues of patch-boundary oscillations or long, stretched triangles in non-uniform rational b-spline surface or triangular mesh representations, respectively. Our algorithm computes appropriate spheroids that are intersected with the c-space to compute the cycloids. We derive a smooth representation of the cycloids using arcs in a rational Bézier formulation. We apply our approach to real-world examples to demonstrate its effectiveness in covering the entire surface with the desired polishing movements.