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The Minkowski sum
(applied to 2d geometry)
cloderic.mars@gmail.com
http://www.crowdscontrol.net
clodericmars
Formal definition
A and B are two sets
A⊕B is the Minkowski sum of A and B
A⊕B = {a+b! a∈A, b∈B}
What if A and B are 2D shapes ?
Hard to visualize ?
Let’s see some examples...
Example 1
A is any polygon
B is a convex polygon
A
B
x
y
A⊕B
x
y
Example 2
A is any polygon
B is any disc
A
B
x
y
A⊕B
x
y
Intuitive definition
What is A⊕B ?
Take B
Dip it into some paint
Put its (0,0) on A border
Translate it along the A
perimeter
The painted area is A⊕B
What can you do with that ?
Notably, motion planning
Free space
A is an obstacle
any 2D polygon
B is a moving object
2D translation : t
shape : a convex polygon or a disc
t ∈ A⊕-B collision
Example 1
A is any polygon
B is a convex polygon
A
B
x
y
-B
A⊕-B
x
y
x
y t
t ∉ A⊕-B no collision
x
y
t
t ∈ A⊕-B collision
Example 2
A is any polygon
B is any disc
A
B=-B
x
y
A⊕-B
x
y
A⊕-B
x
y
t
t ∉ A⊕-B no collision
A⊕-B
x
y
t
t ∈ A⊕-B collision
How is it computed ?
Two convex polygons
ConvexPolygon minkowskiSum(ConvexPolygon a, ConvexPolygon b)
{
Vertex[] computedVertices;
foreach(Vertex vA in a)
{
foreach(Vertex vB in b)
{
computedVertices.push_back(vA+vB);
}
}
return convexHull(computedVertices);
}
Any polygons
Method 1 : decomposition
decompose in convex polygons
compute the sum of each couple
the final sum is the union of each sub-sum
Method 2 : convolution
cf. sources
Polygon offsetting
P is a polygon
D is a disc of radius r
Computing P⊕D = Offsetting P by a radius r
Computation
Easy for a convex polygon
cf. sources
Sources
http://www.cgal.org/Manual/3.4/doc_html/
cgal_manual/Minkowski_sum_2/Chapter_main.html
http://wapedia.mobi/en/Minkowski_addition

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