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Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
Minkowski Sum on 2D geometry
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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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  • 1. The Minkowski sum (applied to 2d geometry) cloderic.mars@gmail.com http://www.crowdscontrol.net clodericmars
  • 2. 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}
  • 3. What if A and B are 2D shapes ? Hard to visualize ? Let’s see some examples...
  • 4. Example 1 A is any polygon B is a convex polygon
  • 5. A B x y
  • 6. A⊕B x y
  • 7. Example 2 A is any polygon B is any disc
  • 8. A B x y
  • 9. A⊕B x y
  • 10. 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
  • 11. What can you do with that ? Notably, motion planning
  • 12. 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
  • 13. Example 1 A is any polygon B is a convex polygon
  • 14. A B x y -B
  • 15. A⊕-B x y
  • 16. x y t t ∉ A⊕-B no collision
  • 17. x y t t ∈ A⊕-B collision
  • 18. Example 2 A is any polygon B is any disc
  • 19. A B=-B x y
  • 20. A⊕-B x y
  • 21. A⊕-B x y t t ∉ A⊕-B no collision
  • 22. A⊕-B x y t t ∈ A⊕-B collision
  • 23. How is it computed ?
  • 24. 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); }
  • 25. 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
  • 26. 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
  • 27. 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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