http://zh.wikipedia.org/wiki/%E9%A4%98%E5%BC%A6%E5%AE%9A%E7%90%86
http://www.cnblogs.com/ch3656468/archive/2011/03/02/1969303.html
http://hi.baidu.com/billdu/blog/item/2b162f7bb799affa2e73b331.html
http://user.qzone.qq.com/122155302/blog/1311607940
http://zh.wikipedia.org/wiki/%E9%A4%98%E5%BC%A6%E5%AE%9A%E7%90%86
方法一：数值积分
除了 Simpson’s rule 以外应该还有其它（哪些？）数值积分的方法，但是由于这方面我的知识不是很牢靠实在是匮乏啊匮乏就没有办法带大家详细讨论了。。（。。。。）
。。 f(x) 表示原函数，s(x) 表示对该函数的不定积分。。
那么 Simpson’s rule 表达除了就是这个。。
.. .
DB s(DB l, DB r){
return (f(l) + 4 * f(m) + f(r)) * (r - l);
}
。。至于 Sevenkplus 所提到的 “自适应 Simpson 公式” 应该就是。。
DB S(DB l, DB r){
DB ss = s(l, r), sl = s(l, m), sr = s(m, r);
if (fabs(ss - sl - sr) < EPS) return sl + sr;
else return S(l, m) + S(m, r);
}
.
也就是设置精度然后微元递降下去。。方法类似以前做的某道 Usaco 几何题。。
下面具体对圆并来说，首先在主程序里去冗余。。也就是忽略退化的圆，然后按半径从小到大排序，标记被整个包含的圆。。（这一步也可以不要。。）
（对于去除被包含这种偏序关系。。如果被包含的个数题目加以限制的话那么可以加入常数优化。。参见 UVa 10902. Pick-up sticks）
之后再按照每个圆左端点的坐标排序，之后是一步扫描线的过程。。得到所有要求的区间。。
最后是圆并里 f(x) 函数的求法。。某横坐标 x 点处 y 轴上被覆盖的面积。。
方法是。。一次环形的扫描线。。然后这里暴力了一点枚举了每一个圆。。。
（。。。好像这里坐标很难离散掉。。于是没有办法用其它什么数据结构。。不知道排序二分会不会有效。）
.. .
vector<pair<DB, DB> > I; // Interval
#define lbd I[i].first
#define rbd I[i].second
DB f(DB xx){
CLR(I); REP(i, n){
DB d = fabs(xx - x[i]);
if (d < r[i]) d = sqrt(sqr(r[i]) - sqr(d)), I.PB(MP(y[i] - d, y[i] + d));
}
SRT(I);
DB Length = 0, ll = -OO, rr = -OO;
REP(i, SZ(I)){
if(rr < lbd) Length += rr - ll, ll = lbd, rr = rbd;
else checkMax(rr, rbd);
}
return Length += rr - ll;
}
.
方法二：Aekdycoin 的方法
。。。核武的方法。。应该就是最标准的扫描线。。因为圆有很多很好的性质，而且再结合有向面积代码量要低于预想。。
首先开局出装和上一种方法一样。。直接进入主要部分。。。
。。枚举每一个圆，计算所有其它圆和它的交点。。进行离散化。。这种环形离散化会产生一些问题。。参见 Master Spark 那支。。
然后大家就都会了吧。
… .. . .. .. .. . ..
。。完整代码。。。。
/** ` Micro Mezzo Macro Flation -- Overheated Economy ., **/
#include <algorithm>
#include <iostream>
#include <iomanip>
#include <sstream>
#include <cstring>
#include <cstdio>
#include <string>
#include <vector>
#include <bitset>
#include <queue>
#include <stack>
#include <cmath>
#include <ctime>
#include <list>
#include <set>
#include <map>
using namespace std;
#define REP(i, n) for (int i=0;i<int(n);++i)
#define FOR(i, a, b) for (int i=int(a);i<int(b);++i)
#define DWN(i, b, a) for (int i=int(b-1);i>=int(a);--i)
#define REP_1(i, n) for (int i=1;i<=int(n);++i)
#define FOR_1(i, a, b) for (int i=int(a);i<=int(b);++i)
#define DWN_1(i, b, a) for (int i=int(b);i>=int(a);--i)
#define REP_C(i, n) for (int n____=int(n),i=0;i<n____;++i)
#define FOR_C(i, a, b) for (int b____=int(b),i=a;i<b____;++i)
#define DWN_C(i, b, a) for (int a____=int(a),i=b-1;i>=a____;--i)
#define REP_N(i, n) for (i=0;i<int(n);++i)
#define FOR_N(i, a, b) for (i=int(a);i<int(b);++i)
#define DWN_N(i, b, a) for (i=int(b-1);i>=int(a);--i)
#define REP_1_C(i, n) for (int n____=int(n),i=1;i<=n____;++i)
#define FOR_1_C(i, a, b) for (int b____=int(b),i=a;i<=b____;++i)
#define DWN_1_C(i, b, a) for (int a____=int(a),i=b;i>=a____;--i)
#define REP_1_N(i, n) for (i=1;i<=int(n);++i)
#define FOR_1_N(i, a, b) for (i=int(a);i<=int(b);++i)
#define DWN_1_N(i, b, a) for (i=int(b);i>=int(a);--i)
#define REP_C_N(i, n) for (n____=int(n),i=0;i<n____;++i)
#define FOR_C_N(i, a, b) for (b____=int(b),i=a;i<b____;++i)
#define DWN_C_N(i, b, a) for (a____=int(a),i=b-1;i>=a____;--i)
#define REP_1_C_N(i, n) for (n____=int(n),i=1;i<=n____;++i)
#define FOR_1_C_N(i, a, b) for (b____=int(b),i=a;i<=b____;++i)
#define DWN_1_C_N(i, b, a) for (a____=int(a),i=b;i>=a____;--i)
#define DO(n) while(n--)
#define DO_C(n) int n____ = n; while(n____--)
#define TO(i, a, b) int s_=a<b?1:-1,b_=b+s_;for(int i=a;i!=b_;i+=s_)
#define TO_1(i, a, b) int s_=a<b?1:-1,b_=b;for(int i=a;i!=b_;i+=s_)
#define SQZ(i, j, a, b) for (int i=int(a),j=int(b)-1;i<j;++i,--j)
#define SQZ_1(i, j, a, b) for (int i=int(a),j=int(b);i<=j;++i,--j)
#define REP_2(i, j, n, m) REP(i, n) REP(j, m)
#define REP_2_1(i, j, n, m) REP_1(i, n) REP_1(j, m)
#define ALL(A) A.begin(), A.end()
#define LLA(A) A.rbegin(), A.rend()
#define CPY(A, B) memcpy(A, B, sizeof(A))
#define INS(A, P, B) A.insert(A.begin() + P, B)
#define ERS(A, P) A.erase(A.begin() + P)
#define BSC(A, X) find(ALL(A), X) // != A.end()
#define CTN(T, x) (T.find(x) != T.end())
#define SZ(A) int(A.size())
#define PB push_back
#define MP(A, B) make_pair(A, B)
#define Rush int T____; RD(T____); DO(T____)
#pragma comment(linker, "/STACK:36777216")
#pragma GCC optimize ("O2")
#define Ruby system("ruby main.rb")
#define Haskell system("runghc main.hs")
#define Pascal system("fpc main.pas")
typedef long long LL;
typedef double DB;
typedef unsigned UINT;
typedef unsigned long long ULL;
typedef vector<int> VI;
typedef vector<char> VC;
typedef vector<string> VS;
typedef vector<LL> VL;
typedef vector<DB> VD;
typedef set<int> SI;
typedef set<string> SS;
typedef set<LL> SL;
typedef set<DB> SD;
typedef map<int, int> MII;
typedef map<string, int> MSI;
typedef map<LL, int> MLI;
typedef map<DB, int> MDI;
typedef map<int, bool> MIB;
typedef map<string, bool> MSB;
typedef map<LL, bool> MLB;
typedef map<DB, bool> MDB;
typedef pair<int, int> PII;
typedef pair<int, bool> PIB;
typedef vector<PII> VII;
typedef vector<VI> VVI;
typedef vector<VII> VVII;
typedef set<PII> SII;
typedef map<PII, int> MPIII;
typedef map<PII, bool> MPIIB;
/** I/O Accelerator **/
/* ... :" We are I/O Accelerator ... Use us at your own risk ;) ... " .. */
template<class T> inline void RD(T &);
template<class T> inline void OT(const T &);
inline int RD(){ int x; RD(x); return x;}
template<class T> inline T& _RD(T &x){ RD(x); return x;}
inline void RC(char &c){scanf(" %c", &c);}
inline void RS(char *s){scanf("%s", s);}
template<class T0, class T1> inline void RD(T0 &x0, T1 &x1){RD(x0), RD(x1);}
template<class T0, class T1, class T2> inline void RD(T0 &x0, T1 &x1, T2 &x2){RD(x0), RD(x1), RD(x2);}
template<class T0, class T1, class T2, class T3> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3){RD(x0), RD(x1), RD(x2), RD(x3);}
template<class T0, class T1, class T2, class T3, class T4> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5, T6 &x6){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5), RD(x6);}
template<class T0, class T1> inline void OT(T0 &x0, T1 &x1){OT(x0), OT(x1);}
template<class T0, class T1, class T2> inline void OT(T0 &x0, T1 &x1, T2 &x2){OT(x0), OT(x1), OT(x2);}
template<class T0, class T1, class T2, class T3> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3){OT(x0), OT(x1), OT(x2), OT(x3);}
template<class T0, class T1, class T2, class T3, class T4> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5, T6 &x6){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5), OT(x6);}
template<class T> inline void RST(T &A){memset(A, 0, sizeof(A));}
template<class T0, class T1> inline void RST(T0 &A0, T1 &A1){RST(A0), RST(A1);}
template<class T0, class T1, class T2> inline void RST(T0 &A0, T1 &A1, T2 &A2){RST(A0), RST(A1), RST(A2);}
template<class T0, class T1, class T2, class T3> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3){RST(A0), RST(A1), RST(A2), RST(A3);}
template<class T0, class T1, class T2, class T3, class T4> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4), RST(A5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4), RST(A5), RST(A6);}
template<class T> inline void CLR(T &A){A.clear();}
template<class T0, class T1> inline void CLR(T0 &A0, T1 &A1){CLR(A0), CLR(A1);}
template<class T0, class T1, class T2> inline void CLR(T0 &A0, T1 &A1, T2 &A2){CLR(A0), CLR(A1), CLR(A2);}
template<class T0, class T1, class T2, class T3> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3){CLR(A0), CLR(A1), CLR(A2), CLR(A3);}
template<class T0, class T1, class T2, class T3, class T4> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4), CLR(A5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4), CLR(A5), CLR(A6);}
template<class T> inline void CLR(T &A, int n){REP(i, n) CLR(A[i]);}
template<class T> inline void FLC(T &A, int x){memset(A, x, sizeof(A));}
template<class T0, class T1> inline void FLC(T0 &A0, T1 &A1, int x){FLC(A0, x), FLC(A1, x);}
template<class T0, class T1, class T2> inline void FLC(T0 &A0, T1 &A1, T2 &A2){FLC(A0), FLC(A1), FLC(A2);}
template<class T0, class T1, class T2, class T3> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3){FLC(A0), FLC(A1), FLC(A2), FLC(A3);}
template<class T0, class T1, class T2, class T3, class T4> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4);}
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4), FLC(A5);}
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4), FLC(A5), FLC(A6);}
template<class T> inline void SRT(T &A){sort(ALL(A));}
template<class T, class C> inline void SRT(T &A, C B){sort(ALL(A), B);}
/** Add - On **/
const int MOD = 1000000007;
const int INF = 0x7fffffff;
const DB PI = acos(-1.0);
const DB EPS = 1e-6;
const DB OO = 1e15;
// <<= ` 0. Daily Use .,
template<class T> inline void checkMin(T &a,const T b){if (b<a) a=b;}
template<class T> inline void checkMax(T &a,const T b){if (b>a) a=b;}
template <class T, class C> inline void checkMin(T& a, const T b, C c){if (c(b,a)) a = b;}
template <class T, class C> inline void checkMax(T& a, const T b, C c){if (c(a,b)) a = b;}
template<class T> inline T min(T a, T b, T c){return min(min(a, b), c);}
template<class T> inline T max(T a, T b, T c){return max(max(a, b), c);}
template<class T> inline T sqr(T a){return a*a;}
template<class T> inline T cub(T a){return a*a*a;}
int Ceil(int x, int y){return (x - 1) / y + 1;}
// <<= ` 1. Bitwise Operation .,
inline bool _1(int x, int i){return x & 1<<i;}
inline int _1(int i){return 1<<i;}
inline int _U(int i){return _1(i) - 1;};
inline int count_bits(int x){
x = (x & 0x55555555) + ((x & 0xaaaaaaaa) >> 1);
x = (x & 0x33333333) + ((x & 0xcccccccc) >> 2);
x = (x & 0x0f0f0f0f) + ((x & 0xf0f0f0f0) >> 4);
x = (x & 0x00ff00ff) + ((x & 0xff00ff00) >> 8);
x = (x & 0x0000ffff) + ((x & 0xffff0000) >> 16);
return x;
}
template<class T> inline T low_bit(T x) {
return x & -x;
}
template<class T> inline T high_bit(T x) {
T p = low_bit(x);
while (p != x) x -= p, p = low_bit(x);
return p;
}
// <<= ` 2. Modular Arithmetic Basic .,
inline void INC(int &a, int b){a += b; if (a >= MOD) a -= MOD;}
inline int sum(int a, int b){a += b; if (a >= MOD) a -= MOD; return a;}
inline void DEC(int &a, int b){a -= b; if (a < 0) a += MOD;}
inline int dff(int a, int b){a -= b; if (a < 0) a += MOD; return a;}
inline void MUL(int &a, int b){a = (LL)a * b % MOD;}
inline int pdt(int a, int b){return (LL)a * b % MOD;}
// <<= ' 0. I/O Accelerator interface .,
template<class T> inline void RD(T &x){
//cin >> x;
scanf("%d", &x);
//char c; for (c = getchar(); c < '0'; c = getchar()); x = c - '0'; for (c = getchar(); c >= '0'; c = getchar()) x = x * 10 + c - '0';
//char c; c = getchar(); x = c - '0'; for (c = getchar(); c >= '0'; c = getchar()) x = x * 10 + c - '0';
}
template<class T> inline void OT(const T &x){
printf("%.3lf\n", x);
}
#define For_each(it, A) for (SII::iterator it = A.begin(); it != A.end(); ++it)
/* .................................................................................................................................. */
inline int sgn(DB x){
return x < -EPS ? -1 : x > EPS;
}
const int N = 1009;
int x[N], y[N], r[N], o[N]; // Circle
#define m ((l + r) / 2)
#define l(a) x[a] - r[a]
#define r(a) x[a] + r[a]
inline bool c1(int a, int b){return r[a] < r[b];} //Sort by Radius
inline bool c2(int a, int b){return l(a) < l(b);} //Sort by left point.
inline bool Cover(int a, int b){return sqr(r[a] - r[b]) >= sqr(x[a] - x[b]) + sqr(y[a] - y[b]);}
int n;
vector<pair<DB, DB> > I; // Interval
map<DB, DB> _f;
#define lbd I[i].first
#define rbd I[i].second
inline DB f(DB xx){
DB &res = _f[xx];
if (res == 0) {
CLR(I); REP(i, n){
DB d = fabs(xx - x[i]);
if (d < r[i]) d = sqrt(sqr(r[i]) - sqr(d)), I.PB(MP(y[i] - d, y[i] + d));
}
SRT(I);
DB ll = -OO, rr = -OO; REP(i, SZ(I)){
if(rr < lbd) res += rr - ll, ll = lbd, rr = rbd;
else checkMax(rr, rbd);
}
res += rr - ll;
}
return res;
}
inline DB s(DB l, DB r){
return (f(l) + 4 * f(m) + f(r)) * (r - l);
}
inline DB _S(DB l, DB r){
DB ss = s(l, r), sl = s(l, m), sr = s(m, r);
if (fabs(ss - sl - sr) < EPS) return sl + sr;
else return _S(l, m) + _S(m, r);
}
inline DB S(DB l, DB r){
CLR(_f); return _S(l, r);
}
int main(){
//freopen("in.txt", "r", stdin);
RD(n); for (int i=0;i<n;r[i]?++i:--n) RD(x[i], y[i], r[i]);
// Delete some useless circle .. .
REP(i, n) o[i] = i; sort(o, o + n, c1);
REP(i, n) FOR(j, i+1, n) if (Cover(o[j], o[i])){o[i] = -1; break;}
int _n = n; n = 0; REP(i, _n) if (o[i] != -1) o[n++] = o[i];
sort(o, o + n, c2);
int _x[N], _y[N], _r[N];
REP(i, n) _x[i] = x[o[i]], _y[i] = y[o[i]], _r[i] = r[o[i]];
CPY(x, _x), CPY(y, _y), CPY(r, _r);
// Calculate Area by the Simpson's Rule .. .
DB Area = 0, ll = -OO, rr = -OO;
REP(i, n){
if(rr < l(i)) Area += S(ll, rr), ll = l(i), rr = r(i);
else checkMax(rr, DB(r(i)));
}
Area += S(ll, rr), Area /= 6;
OT(Area);
}
