# Nonconvexity, and playing indoor paintball

Following the two previous posts (here and there), on the number of people that don’t get wet while playing with water pistols, consider now an indoor version, in a non-convex room (i.e. player behind wall are now, somehow, protected). In the previous posts, players where playing on a square field, and I briefly mentioned that if the field was a disk, results would have been (roughly) the same: so far, the shape of the field was not an issue. But what if the field is no longer convex,

```library(sp)
plot(0:2,0:2,col="white",xlab="",ylab="")
MAP=Polygon(cbind(c(0,0,1,1,2,2,0),
c(0,2,2,1,1,0,0)))
polygon(MAP@coords,col="light blue")```

and players hidden behind the wall cannot be reached (red lines above are impossible hits). As earlier, it is still possible to look at the closest neighbor, we just have to exclude pairs that can no longer hit each other.

And again, it is possible to plot safe zones in green.

Once again, it is possible to look more closely are those supposed-to-be “safe zones”, i.e. by looking at the distribution of the location of players that were dry at the end of the game. With 11 players, we obtain

What about the distribution of the number of dry players, over a game ?

```touch=function(x1,y1,x2,y2,n=251){
X=seq(x1,x2,length=n)
Y=seq(y1,y2,length=n)
sum(point.in.polygon(X,Y,MAP@coords[,1],
MAP@coords[,2], mode.checked=FALSE)==0)==0
}

NOTWETnc=function(n,p){
sx=runif(50)*2;sy=runif(50)*2
IN=which(point.in.polygon(sx,sy,MAP@coords[,1],
MAP@coords[,2], mode.checked=FALSE)==1)
Sx=sx[IN];Sy=sy[IN]
Sx=Sx[1:n];Sy=Sy[1:n]
IN=IN[1:n]
MI=matrix(NA,n,n)
for(i in 1:(n-1)){
for(j in (i+1):(n)){
MI[j,i]=MI[i,j]=touch(Sx[i],Sy[i],Sx[j],Sy[j])
}}
(d=as.matrix(dist(cbind(Sx,Sy),
method = "euclidean",upper=TRUE)))
diag(d)=999999
dpossible=d
dpossible[MI==FALSE]=999999
dmin=apply(dpossible,2,which.min)
#whonotwet=( (1:n) %notin% names(table(dmin)) )
notwet=n-length(table(dmin))
return(notwet)}

NOTWET=function(n){
x=runif(n)
y=runif(n)
(d=as.matrix(dist(cbind(x,y),
method = "euclidean",upper=TRUE)))
diag(d)=999999
dmin=apply(d,2,which.min)
notwet=n-length(table(dmin))
return(notwet)}

NSim=10000
Nnc=Vectorize(NOTWETnc)(n=rep(11,NSim))
Nc=Vectorize(NOTWET)(n=rep(11,NSim))
T=table(Nc)
Tn=table(Nnc)
plot(as.numeric(names(Tn)),
Tn/NSim,type="b",col="blue")
lines(as.numeric(names(T)),
T/NSim,type="b",col="red",pch=4)```

On 11 players, we have the same distribution as the one on a square field. So convexity is not a key issue here…

Strange isn’t it. And with an odd number of player, not only there is at least one dry player, but at least, half of the players (maybe minus one) have to be wet…

Cite this blog post
Arthur Charpentier (2012, April 6). Nonconvexity, and playing indoor paintball. Freakonometrics. Retrieved June 23, 2024, from https://doi.org/10.58079/oul2

## 3 thoughts on “Nonconvexity, and playing indoor paintball”

1. Robes de soirée roses says:

great blog, I really enjoy reading it.

2. thanks, but actually, it looks like that binomial distribution does not depend at all on the shape of the field, .e.g.

` library(sp) plot(0:2,0:2,col="white",xlab="",ylab="") MAP=Polygon(cbind(c(0,0,.75,.75,1.25, 1.25,2,2,1.25,1.25,.75,.75,0), c(.75,1.25,1.25,2,2,1.25,1.25, .75,.75,0,0,.75,.75))) polygon(MAP@coords,col="light blue")`

..

here we have, again, the same distribution for the number of dry players,

3. greg says:

great seriies of posts, and great blog, I really enjoy reading it. On this specific post, you have to admit that your battlefield is not that convex. Will the distribution change if you consider something more convex, like a star, or a cross, at least

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