jzoj6678 苏菲的世界 (辛普森积分，圆与圆并)

题意

20个球，求面积并

做法

辛普森积分+圆并。
辛普森积分：知道一个函数在$l,r,0.5(l+r)$处的值，可以估计其区间积分为$(r-l)\frac{(f(l)+f(r)+4f(0.5(l+r)))}{6}$。二分处理，直到$r-l<=gap$且当前区间的估计值与子区间的估计值之差小于eps为止。
圆并做法：先去重合，去包含，然后考虑上轮廓减下轮廓，枚举每一个圆，用一些几何求出所有被覆盖的区间，然后再对没有被覆盖的区间做积分。

#include 
using namespace std;
typedef double db;
const db gap = 0.1, pi = acos(-1), eps = 1e-6;
const int N = 30;
int n, x[N], y[N], z[N], r[N];
struct pot{
	db x,y;
	pot(db _x=0, db _y=0) {x=_x,y=_y;}
	pot operator-(pot b){return pot(x-b.x,y-b.y);}
	pot operator+(pot b){return pot(x+b.x,y+b.y);}
	db operator*(pot b){return x*b.y-y*b.x;}
	pot operator*(db r){return pot(x*r,y*r);}
} o[N];
db R[N];
int m, del[N];
#define sqr(x) ((x)*(x))
#define zero(x) (abs(x)<=eps)
db len2(pot a) {
	return sqr(a.x)+sqr(a.y);
}

pair<db,int> tp[N * 2];
int cc = 0;
void addseg(db l, db r) {
	tp[++cc]=make_pair(l,1);
	tp[++cc]=make_pair(r,-1);
}

db ciruni() {
	db ret = 0;
	for(int i = 1; i <= m; i++) {
		cc = 0;
		tp[++cc]=make_pair(0,0);
		tp[++cc]=make_pair(-pi,0);
		tp[++cc]=make_pair(pi,0);
		for(int j = 1; j <= m; j++) if (i != j) {
			db d2 = len2(o[i] - o[j]);
			db d = sqrt(d2);
			if (d >= R[i] + R[j]) continue;
			db cs = (sqr(R[i]) + d2 - sqr(R[j])) * 0.5 / (R[i] * d);
			db ang = acos(cs);
			db b = atan2(o[j].y - o[i].y, o[j].x - o[i].x);
			db l = b - ang, r = b + ang;
			if (-pi <= l && r <= pi)addseg(l,r);
			else if (r > pi) {
				addseg(l,pi), addseg(-pi,r-2*pi);
			} else if (l < -pi) {
				addseg(-pi,r),addseg(l+2*pi,pi);
			}
		}
		sort(tp+1,tp+1+cc);
		int js=0;
		for(int j=1;j<cc;j++){
			js += tp[j].second;
			if(!zero(tp[j].first - tp[j + 1].first) && js == 0) {
				db jl = tp[j].first, jr = tp[j + 1].first;
				pot l = o[i] + pot(cos(jl), sin(jl)) * R[i];
				pot r = o[i] + pot(cos(jr), sin(jr)) * R[i];
				db w = jr - jl;
				ret -= (l.y + r.y) * 0.5 * (r.x - l.x);
				ret += w * R[i] * R[i] * 0.5 - R[i] * R[i] * sin(w) * 0.5;
			}
		}
	}
	return ret;
}

db calc_z(db tz) {
	m=0;
	db xmx=-1000,xmi=1000;
	for(int i = 1; i <= n; i++) {
		if (z[i] - r[i] <= tz && tz <= z[i] + r[i]) {
			++m;
			o[m]=pot(x[i],y[i]);
			R[m]=sqrt(r[i]*r[i]-sqr(tz-z[i]));
			xmx=max(xmx,x[i]+R[m]);
			xmi=min(xmi,x[i]-R[m]);
		}
	}
	if (m==0)return 0;
	if (m==1)return pi*R[1]*R[1];
	memset(del,0,sizeof del);
	for(int i = 1; i <= m; i++) {
		for(int j = i + 1; j <= m; j++) {
			if (zero(o[i].x-o[j].x) && zero(o[i].y-o[j].y) && zero(R[i]-R[j])) {
				del[j] = 1;
			}
		}
	}
	for(int i=1;i<=m;i++){
		for(int j=1;j<=m;j++)if(j!=i&&R[i]>R[j]){
			if (len2(o[i]-o[j])<=sqr(R[i]-R[j]))del[j]=1;
		}
	}
	int tm=0;
	for(int i=1;i<=m;i++)if(!del[i]){
		o[++tm]=o[i];
		R[tm]=R[i];
	}
	m=tm;
	db v = ciruni();
//	printf("%.5lf %.5lf\n",tz,v);
	return v;
}

db simp_z(db l, db r, db fl, db fr, db &fmid) {
	return (r-l)/6*(fl+fr+4*(fmid=calc_z((l+r)*0.5)));
}

db sum_z(db lz, db rz, db s, db fl, db fr, db fmid) {
	db mid = 0.5 * (lz + rz), lmid = 0, rmid = 0;
	db ansl = simp_z(lz, mid, fl, fmid, lmid);
	db ansr = simp_z(mid, rz, fmid, fr, rmid);
	if (rz - lz > gap || abs(ansl + ansr - s) >= eps) {
		return sum_z(lz, mid, ansl, fl, fmid, lmid) + sum_z(mid, rz, ansr, fmid, fr, rmid);
	} else return ansl + ansr;
}

db start_z(db lz, db rz) {
	db fl = calc_z(lz), fr = calc_z(rz), fmid = 0, z = simp_z(lz, rz, fl, fr, fmid);
	return sum_z(lz, rz, z, fl, fr, fmid);
}

int main() {
	freopen("sphere.in","r",stdin);
	freopen("sphere.out","w",stdout);
	cin >> n;
	int zmx=-1000,zmi=1000;
	for(int i=1;i<=n;i++){
		scanf("%d %d %d %d",&x[i],&y[i],&z[i],&r[i]);
		zmx=max(zmx,z[i]+r[i]);
		zmi=min(zmi,z[i]-r[i]);
	}
	db ans = start_z(zmi, zmx);
	printf("%.3lf\n",ans);
}