Minkowski addition
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In geometry, the Minkowski sum — also known as dilation — of two sets A and B in Euclidean space is the result of adding every element of A to every element of B, i.e. the set
For example, if we have two 2-simplices (triangles), with points represented by
- A = { (1, 0), (0, 1), (0, −1)}
and
- B = { (0, 0), (1, 1), (1, −1)},
then the Minkowski sum is
- A + B = { (1, 0), (2, 1), (2, −1), (0, 1), (1, 2), (1, 0), (0, −1), (1, 0), (1, −2)}, which looks like a hexagon, with three 'repeated' points at (1,0).
This defines a binary operation called Minkowski addition, named after Hermann Minkowski. It occurs in a basic step in proving Minkowski's theorem, in the form
- C + C = 2C
for a convex symmetric set containing 0, where the left-hand side is the Minkowski sum and the right-hand side the enlargement by a factor of 2.
This operation is sometimes called (somewhat inappropriately) the convolution of the two sets. The actual convolution of the indicator functions of the set will be a function with the same support as the Minkowski sum.
Minkowski addition is also called the binary dilation of A by B.
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[edit] Essential Minkowski sum
There is also a notion of the essential Minkowski sum +e of two subsets of Eulidean space. Note that the usual Minkowski sum can be written as
Thus, the essential Minkowski sum is defined by
where μ denotes n-dimensional Lebesgue measure. The reason for the term "essential" is the following property of indicator functions: while
it can be seen that
where ess sup denotes the essential supremum.
[edit] Applications
Minkowski addition plays a central role in mathematical morphology. It arises in the brush-and-stroke paradigm of 2D computer graphics (pioneered by Donald E. Knuth in Metafont), and as the solid sweep operation of 3D computer graphics.
[edit] Motion planning
Minkowski sums are used in motion planning of an object among obstacles. they are used for the computation of the configuration space, which is the set of all admissible positions of the object. In the simple model of translational motion of an object in the plane, where the position of an object may be uniquely specified by the position of a fixed point of this object, the configuration space are the Minkowski sum of the set of obstacles and the movable object.
[edit] NC machining
In NC machining, the programming of the NC tool exploits the fact that the Minkowski sum of the cutting piece with its trajectory gives the shape of the cut in the material.
[edit] Algorithms for computing Minkowski sums
[edit] Planar case
[edit] Two convex polygons in the plane
For two convex polygons P and Q in the plane with m and n vertices, their Minkowski sum is a convex polygon m + n vertices and may be computed in time O (m + n) by a very simple procedure, which may be informally described as follows. Assume that the edges of a polygon are given the direction, say, counterclockwise, along the polygon boundary. Then it is easily seen that these edges of the convex polygon are ordered by polar angle. Let us merge the ordered sequences of the directed edges from P and Q into a single ordered sequence S. Imagine that these edges are solid arrows which can be moved freely while keeping them parallel to their original direction. Assemble these arrows in the order of the sequence S by attaching the tail of the next arrow to the head of the previous arrow. It turns out that the resulting polygonal chain will in fact be a convex polygon which is the Minkowski sum of P and Q.
[edit] Other
If one polygon is convex and another one is not, the complexity of their Minkowski sum is O(nm). If both of them are nonconvex, their Minkowski sum complexity is O((mn)2)
[edit] See also
[edit] References
- Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.) 39 (3): 355–405 (electronic).
