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Minkowski sums are of theoretical interest and have applications in fields related to industrial backgrounds. In this paper we focus on the specific case of summing polytopes as we want to solve the tolerance analysis problem described in [1]. Our approach is based on the use of linear programming and is solvable in polynomial time. The algorithm we developed can be implemented and parallelized in a very easy way.

Tolerance analysis is the branch of mechanical design dedicated to studying the impact of the manufacturing tolerances on the functional constraints of any mechanical system. Minkowski sums of polytopes are useful to model the cumulative stack-up of the pieces and thus, to check whether the final assembly respects such constraints or not, see [

Given two sets

A polytope is defined as the convex hull of a finite set of points, called the

In this paper we deal with

Let

We recall that in [

Reciprocally let

Let

We know that

The reciprocal is obvious as

At this step an algorithm removing all points which are not vertices of

To perform such a task, a popular technique given in [

So if we define the matrix

then

The corresponding method is detailed in Algorithm 2. Now we would like to find a way to reduce the size of the main matrix

In this section we want to use the basic property 1 characterizing a Minkowski vertex. Then the algorithm computes, as done before, all sums of pairs

We get the following system:

That is to say with matrices and under the hypothesis of positivity for both vectors

We are not in the case of the linear feasibility problem as there is at least one obvious solution:

The question is to know whether it is unique or not. This first solution is a vertex

So we can write it in a more general form:

where only the second member is function of

It gives the linear programming system:

Thanks to this system we have now the basic property the algorithm relies on:

It is also interesting to note that when the maximum

The current state of the art runs

Let’s switch now to the geometric interpretation, given

The method tries to build a pair, if it exists,

So the question about

The existence of a straight line inside the reunion of the cones is equivalent to the existence of a pair

We can resume the property writing it as an intersection introducing the cone

In this paper, our algorithm goes beyond the scope of simply finding the vertices of a cloud of points. That’s why we have characterized the Minkowski vertices. However, among all the properties, some of them are not easily exploitable in an algorithm. In all the cases we have worked directly in the polytopes

of property 6 where the intersection is computed with primal cones. It actually implements Weibel’s approach described in [

We would like to thank Pr Pierre Calka from the LMRS in Rouen University for his precious help in writing this article.

Vincent Delos,Denis Teissandier, (2015) Minkowski Sum of Polytopes Defined by Their Vertices. Journal of Applied Mathematics and Physics,03,62-67. doi: 10.4236/jamp.2015.31008