Algorithm to calculate the Minkowski sums of 3-polytopes : application to tolerance analysis

In tolerance analysis, it is necessary to check that the cumulative defect limits specified for the component parts of a product are compliant with the functional requirements expected of the product. Defect limits can be modelled by tolerance zones constructed by offsets on nominal models of parts. Cumulative defect limits can be modelled using a calculated polytope, the result of a set of intersections and Minkowski sums of polytopes. A functional requirement can be qualified by a functional polytope, in other words a target polytope. It is then necessary to verify whether the calculated polytope is included in the functional polytope. The purpose of this article is to determine the Minkowski sum of 3-dimension polytopes and apply this effectively in order to optimise the filling of the functional polytope. This method can be used to determine from which vertices of the operands the vertices of the Minkowski sum derive. It is also possible to determine to which facets of the operands each facet of the Minkowski sum is oriented. First, the main properties of the duality of normal cones and primal cones associated with the vertices of a polytope are described. Next, the properties of normal fans are applied to define the vertices and facets of the Minkowski sum of two polytopes. An algorithm is proposed which generalises the method. Lastly, there is a discussion of the features of this algorithm, developed using the OpenCascade environment