sborník

25. KONFERENCE O GEOMETRII A POČÍTAČOVÉ GRAFICE
Světlana Tomiczková
AREA OF THE MINKOWSKI SUM OF
TWO CONVEX SETS
Abstract
This paper deals with the area of the Minkowski sum of two
convex polygons and the area of the Minkowski sum of two
convex sets bounded by closed curves.
Minkowski sum, area.
Keywords
1 Introduction
The Minkowski sum of two point sets A and B can be defined as
A ⊕ B = ⋃ A b or A ⊕ B = {a + b| a ∈ A ∧ b ∈ B}.
b∈B
We will find the rule for the computation of the area of the
Minkowski sum of two convex sets.
2 Area of the Minkowski sum of two convex
polygons
Theorem 1: (see fig. 1) Let A and B be convex polygons and
O = [0, 0] an inner point of the polygon B and C = A ⊕ B. Then the
area S(C) of the polygon C is
S(C) = S(A) + S(B) +
n∑
|a i ||v i |,
where S(A), S(B) are areas of the polygons A, B, a i is the edge of the
polygon A, n i is the outer normal vector of a i and v i is the distance of
the extreme point in direction n i on the polygon B from the straight
line which is parallel with a i and goes through the point O = [0, 0].
Proof: (see fig. 2) Let A 1 , . . . , A n be the vertices of the polygon
A and B 1 , . . . , B m the vertices of the polygon B. We denote the
i=1
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