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.