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Math Help - Hausdorf distance

  1. #1
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    Hausdorf distance

    How can I determine the Hausdorf distance between 2 sets when the intersection of those sets is not empty?

    For example,

    U = \{ (x,y) : \, 1 \leq x \leq 4 \text{ and } 1 \leq y \leq 2 \}
    V = \{ (x,y) : \, 1 \leq x \leq 2 \text{ and } 0 \leq y \leq 2 \}

    Can someone help?

    QT \pi
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  2. #2
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    Quote Originally Posted by qtpi View Post
    How can I determine the Hausdorf distance between 2 sets when the intersection of those sets is not empty?

    For example,

    U = \{ (x,y) : \, 1 \leq x \leq 4 \text{ and } 1 \leq y \leq 2 \}
    V = \{ (x,y) : \, 1 \leq x \leq 2 \text{ and } 0 \leq y \leq 2 \}

    Can someone help?

    QT \pi
    The definition of the Hausdorff distance between U and V is

    \displaystyle d_H(U,V) = \max\{\sup_{u\in U}\inf_{v\in V}d(u,v),\sup_{v\in V}\inf_{u\in U}d(u,v)\}.

    In everyday language, what that means is this. You look for the furthest distance from V that a point of U can be. Then you look for the furthest distance from U that a point of V can be. Finally, you take the larger of those two distances.

    \setlength{\unitlength}{10mm}<br />
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\put(0,1){\line(1,0){6}}<br />
\thicklines<br />
\put(2,1){\line(1,0){1}}<br />
\put(2,2){\line(1,0){3}}<br />
\put(2,3){\line(1,0){3}}<br />
\put(2,1){\line(0,1){2}}<br />
\put(3,1){\line(0,1){2}}<br />
\put(5,2){\line(0,1){1}}<br />
\put(1.9,0.5){$1$}<br />
\put(2.9,0.5){$2$}<br />
\put(4.9,0.5){$4$}<br />
\put(0.7,0.6){$0$}<br />
\put(0.7,1.8){$1$}<br />
\put(0.7,2.8){$2$}<br />
\put(2.4,1.4){$V$}<br />
\put(3.5,2.4){$U$}<br />
\end{picture}

    For the example in the picture, the furthest distance from V that you can get while remaining in U is 2 (all the points on the right-hand edge of U are distance 2 from V). The furthest distance from U that you can get while remaining in V is 1 (all the points on the bottom edge of V are distance 1 from U). The maximum of the numbers 2 and 1 is 2. Therefore d_H(U,V) = 2.

    In that calculation, it makes no difference whether or not the intersection of U and V is empty.
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