sborník

AREA OF THE MINKOWSKI SUM
Figure 2: Area of the Minkowski sum of two
convex polygons 2
part of C.
1. There is an edge c i k to each edge a i such that a i ‖ c i k and
|a i | = |c i k |. We can obtain ci k by moving the edge a i by the
position vector b j of the extreme point B j in direction n i on
the polygon B and C k = A i + b j .
We can create the parallelogram R i to each edge a i such that
the two edges of R i are a i = A i A i+1 and c i k = C kC k+1 (a i ‖
c i k
). The remaining two edges are equivalent and parallel to the
position vectors of the extreme point B j in direction n i on the
polygon B. The area of the parallelogram R i is S(R i ) = |a i ||v i |
where v i is the altitude of the parallelogram R i to the edge a i .
It is the same as the distance of the extreme vertex in direction
n i on the polygon B from the straight line which is parallel to
the edge a i and goes through the point O = [0, 0].
2. The edges c j k = C kC k+1 are parallel and equivalent to the edges
of the polygon B. Since C k = A i + b j a C k+1 = A i + b j+1 we
can create the triangle T j = C k C k+1 A i to each edge c j k . The
triangle T j is equivalent to the triangle OB j B j+1 . The sum
of the areas of these triangles gives the area of the polygon B.
Thus ∑ m
j=1 S(T j) = S(B).
The polygon A, the parallelograms R i and triangles T j fill the
whole polygon C because the only situations that can arise are as
257