A model for predicting the shape of the cut profile in industrial cutting processes has been developed and applied to abrasive water jet cutting. The well-known physics of abrasive wear is used in the process model, which is cast in intrinsic coordinates. This choice of coordinates enables a formulation of the problem that leads to simple solutions and also a good understanding of the process. The model predicts that fluctuations in the local curvature of the cutting face are an inherent property of the cutting process and that no matter how precise the control over the cutting parameters, the fluctuations cannot be eliminated. Data from cuts filmed whilst cutting Perspex have been used to validate the model.