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Minkowski Sum on 2D geometry

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A simple presentation of a the Minkowski Sum applied to 2D geometry.

A simple presentation of a the Minkowski Sum applied to 2D geometry.

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    Minkowski Sum on 2D geometry Minkowski Sum on 2D geometry Presentation Transcript

    • The Minkowski sum(applied to 2d geometry)cloderic.mars@gmail.comhttp://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 1A is any polygonB is a convex polygon
    • ABy x
    • A⊕By x
    • Example 2A is any polygonB is any disc
    • A By x
    • A⊕By x
    • 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 spaceA is an obstacle any 2D polygonB is a moving object 2D translation : t shape : a convex polygon or a disc t ∈ A⊕-B collision
    • Example 1A is any polygonB is a convex polygon
    • ABy x -B
    • A⊕-B y x
    • y t x t ∉ A⊕-B no collision
    • y t x t ∈ A⊕-B collision
    • Example 2A is any polygonB is any disc
    • AB=-By x
    • A⊕-By x
    • A⊕-By t t ∉ A⊕-B no collision x
    • A⊕-By t t ∈ A⊕-B collision x
    • How is it computed ?
    • Two convex polygonsConvexPolygon 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 polygonsMethod 1 : decomposition decompose in convex polygons compute the sum of each couple the final sum is the union of each sub-sumMethod 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
    • Sourceshttp://www.cgal.org/Manual/3.4/doc_html/cgal_manual/Minkowski_sum_2/Chapter_main.htmlhttp://wapedia.mobi/en/Minkowski_addition