       Re: Distance from point to set

• To: mathgroup at smc.vnet.net
• Subject: [mg55329] Re: [mg55273] Distance from point to set
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sat, 19 Mar 2005 04:45:50 -0500 (EST)
• References: <200503181033.FAA14616@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```On 18 Mar 2005, at 11:33, Piotr Kowalski wrote:

> Hello,
>
> I would like to compute distance d(x,A) from a point 'x' to a set 'A',
> (all in R^n, where n=2 or n=3) that is:
>
>    d(x,A) = min ||x - a|| (forall a in A)
>    where: n=2 or n=3,
>           x is point in R^n, A is subset of R^n
>           ||  || is norm (euclidean, max, etc).
>
> Can I find Mathematica function or package for such problem ?
>
> Thank you in advance,
> P. Kowalski
>

If you have Mathematica 5 or above you can use simply:

dist[p_, l_, n_:2] := Min[Norm[#, n] & /@ Transpose[Transpose[l] - p]]

For example, let

In:=
p = {x, y}; l = {{a, b}, {c, d}};

then

In:=
dist[p, l]

Out=
Min[Sqrt[Abs[a - x]^2 + Abs[b - y]^2],
Sqrt[Abs[c - x]^2 + Abs[d - y]^2]]

In:=
dist[p, l, 4]

Out=
Min[(Abs[a - x]^4 + Abs[b - y]^4)^(1/4),
(Abs[c - x]^4 + Abs[d - y]^4)^(1/4)]

In:=
dist[p, l, Infinity]

Out=
Min[Max[Abs[a - x], Abs[b - y]], Max[Abs[c - x],
Abs[d - y]]]

Andrzej Kozlowski
Chiba, Japan
http://www.akikoz.net/andrzej/index.html
http://www.mimuw.edu.pl/~akoz/

```

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