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The surface-trajectory model gives a solution to some of the problems presented by the general geometric models where the design of an object is separated from its manufacture. In fact, in this model the internal representation of objects is made up of machining trajectories. As the display systems usually need triangles to represent the objects, a process of triangulation is needed to visualize them. In other words, a secondary surface model is needed to display the objects. The following is a geometric model that, maintaining the philosophy of the surface-trajectory model encapsulates the calculation of the machining process from the formal framework that provides the set theory and the mathematical morphology. The model addresses the process of designing objects by assimilation of a machining process by giving solutions to the design of complex objects and an arithmetic to support the generation of trajectories of manufacturing. The design process is similar to the craft work of sculptors designing their pieces by hand with their tools. It also gives a direct solution to the problems of the trajectory generation since they are already defined at the design phase. The model is generic and robust as there are no special cases or complex objects in which the model does not provide a correct solution. It also naturally incorporates the realistic display of the machined objects in a quickly and accurately way. [DOI: 10.1115/1.4002193] Keywords: mathematical morphology, machining, geometric model

Background. Tool path generation problem is one of the most complexes in computer aided manufacturing. Although some efficient algorithms have been developed to solve it, their technological dependency makes them efficient in only a limited number of cases. Method of Approach. Our aim is to propose a model that will set apart the geometrical issues involved in the manufacturing process from the purely technology-dependent physical issues by means of a topological system. This system applies methods and concepts used in mathematical morphology paradigms. Thus, we will obtain a geometrical abstraction which will not only provide solutions to typically complex problems but also the possibility of applying these solutions to any machining environment regardless of the technology; Presented in the paper is a method for offsetting any kind of curve. Specifically, we use parametric cubic curves, which is one of the most general and popular models in computer aided design (CAD)/computer aided manufacturing (CAM) applications. Results. The resulting method avoids any constraint in object or tool shape and obtains valid and optimal trajectories, with a low temporal cost of O(n . m), which is corroborated by the experiments. It also avoids some precision errors that are present in the most popular commercial CAD/CAM libraries. Conclusions. The use of morphology as the base of the formulation avoids self-intersections and discontinuities and allows the system to machine free-form shapes using any tool without constraints. Most numerical and geometrical problems are also avoided. Obtaining a practical algorithm from the theoretical formulation is straightforward. The resulting procedure is simple and efficient. [DOI: 10.111511.2540741] Keywords: offset, contour pocketing, mathematical morphology