Hausdorff Distance Between Convex PolygonsI'm interested in calculating the Hausdorff Distance between 2 polygons (specifically quadrilaterals which are almost rectangles) defined by their vertices. They may overlap.Recall d H ( A , B ) = max ( d ( A , B ) , d ( B , A ) ) where d is the Hausdorff semi-metric d ( A , B ) = sup a ∈ A inf b ∈ B d ( a , b ).Is it true that, given a finite disjoint covering of A, { A i }, d ( A , B ) = max { d ( A i , B ) }? A corollary of which is that d ( A , B ) = d ( A ∖ B , B ).

Question

Hausdorff Distance Between Convex PolygonsI&#039;m interested in calculating the Hausdorff Distance between 2 polygons (specifically quadrilaterals which are almost rectangles) defined by their vertices. They may overlap.Recall       d    H    (  A  ,  B  )  =  max  (  d  (  A  ,  B  )  ,  d  (  B  ,  A  )  ) where d is the Hausdorff semi-metric   d  (  A  ,  B  )  =      sup          a      ∈      A            inf          b      ∈      B        d  (  a  ,  b  ).Is it true that, given a finite disjoint covering of A,   {      A    i    },   d  (  A  ,  B  )  =  max  {  d  (      A    i    ,  B  )  }? A corollary of which is that   d  (  A  ,  B  )  =  d  (  A  ∖  B  ,  B  ).

Kennedy Evans · Accepted Answer

Explanation:Yes, if   ∪      A    i    =  A. This follows from transitivity of comparison operators. It does not matter how you divide your points,   max  {  sup  A  ,  sup  B  }  =  sup  A  ∪  B

I'm interested in calculating the Hausdorff Distance between 2 polygons (specifically quadrilaterals which are almost rectangles) defined by their vertices. They may overlap. Recall d_H(A, B)=max(d(A, B),d(B, A)) where d is the Hausdorff semi-metric d(A, B)=sup_{a in A} inf_{b in B}d(a,b).

Answered question

Answer & Explanation

New Questions in High school geometry