n^2枚举圆盘，用两圆圆心的向量的极角+余弦定理求某个圆覆盖了该圆的哪一段区间（用弧度表示），最后求个区间并。
注意……精度……最好再累计区间的时候，把每个区间的长度减去EPS，防止最后覆盖的总区间超过2PI一点点，使答案为负。
#include<cstdio>
#include<cmath>
#include<algorithm>
using namespace std;
#define EPS 0.000000001
#define INF 1000000.0
int n;
const double PI=acos(-1.0);
double rs[1001],xs[1001],ys[1001],ans;
struct Seg
{
double l,r;
Seg(){}
Seg(const double &L,const double &R){l=L; r=R;}
}q[10001];
bool cmp(const Seg &a,const Seg &b)
{
return fabs(a.l-b.l)<EPS ? a.r<b.r : a.l<b.l;
}
double sqr(const double &x)
{
return x*x;
}
double dis(double x1,double y1,double x2,double y2)
{
return sqrt(sqr(x1-x2)+sqr(y1-y2));
}
int main()
{
// freopen("bzoj1043.in","r",stdin);
scanf("%d",&n);
for(int i=1;i<=n;++i)
scanf("%lf%lf%lf",&rs[i],&xs[i],&ys[i]);
for(int i=1;i<=n;++i)
{
int e=0;
double Lef=2.0*PI;
for(int j=i+1;j<=n;++j)
{
double d=dis(xs[i],ys[i],xs[j],ys[j]);
if(d+rs[i]-rs[j]<EPS)
{
Lef=0.0;
break;
}
else if(d+rs[j]-rs[i]<EPS || rs[i]+rs[j]-d<-EPS)
continue;
// double see=(sqr(d)+sqr(rs[i])-sqr(rs[j]))*0.5/d/rs[i];
double jiao2=acos((sqr(d)+sqr(rs[i])-sqr(rs[j]))*0.5/d/rs[i]);
double jiao1=atan2(ys[j]-ys[i],xs[j]-xs[i]);
if(jiao1<-EPS)
jiao1+=(2.0*PI);
if(jiao1-jiao2<-EPS)
{
q[++e]=Seg(0.0,jiao1+jiao2);
q[++e]=Seg(jiao1-jiao2+2.0*PI,2.0*PI);
}
else if(jiao1+jiao2-2.0*PI>EPS)
{
q[++e]=Seg(jiao1-jiao2,2.0*PI);
q[++e]=Seg(0.0,jiao1+jiao2-2.0*PI);
}
else
q[++e]=Seg(jiao1-jiao2,jiao1+jiao2);
}
// for(int j=e;j;--j)
// {
// bool fl=0;
// for(int k=j-1;k;--k)
// if(!(q[j].l-q[k].r>EPS||q[k].l-q[j].r>EPS))
// {
// q[k].l=min(q[j].l,q[k].l);
// q[k].r=max(q[j].r,q[k].r);
// fl=1;
// break;
// }
// if(!fl)
// Lef-=(q[j].r-q[j].l);
// }
Seg now=Seg(INF,INF);
sort(q+1,q+e+1,cmp);
for(int j=1;j<=e;++j)
if(j==e || q[j+1].l-q[j].l>EPS)
{
if(fabs(now.l-INF)<EPS) now=q[j];
else
{
if(q[j].l-now.r<EPS) now.r=max(now.r,q[j].r);
else
{
Lef-=(now.r-now.l);
now=q[j];
}
}
}
if(fabs(now.l-INF)>=EPS)
Lef-=(now.r-now.l);
if(Lef>EPS)
ans+=(Lef*rs[i]);
}
printf("%.3lf\n",ans);
return 0;
}