圓並 .. .

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:數值積分

除了 Simpson’s rule 以外應該還有其它(哪些?)數值積分的方法,但是由於這方面我的知識不是很牢靠實在是匱乏啊匱乏就沒有辦法帶大家詳細討論了。。(。。。。)

。。 f(x) 表示原函數,s(x) 表示對該函數的不定積分。。
那麼 Simpson’s rule 表達除了就是這個。。

.. .
DB s(DB l, DB r){
    return (f(l) + 4 * f(m) + f(r)) * (r - l);
}

。。至於 Sevenkplus 所提到的 「自適應 Simpson 公式」 應該就是。。

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);
}
.

也就是設置精度然後微元遞降下去。。方法類似以前做的某道 Usaco 幾何題。。
下面具體對圓並來說,首先在主程序里去冗餘。。也就是忽略退化的圓,然後按半徑從小到大排序,標記被整個包含的圓。。(這一步也可以不要。。)

(對於去除被包含這種偏序關係。。如果被包含的個數題目加以限制的話那麼可以加入常數優化。。參見 UVa 10902. Pick-up sticks

之後再按照每個圓左端點的坐標排序,之後是一步掃描線的過程。。得到所有要求的區間。。
最後是圓並里 f(x) 函數的求法。。某橫坐標 x 點處 y 軸上被覆蓋的面積。。

方法是。。一次環形的掃描線。。然後這裡暴力了一點枚舉了每一個圓。。。
(。。。好像這裡坐標很難離散掉。。於是沒有辦法用其它什麼數據結構。。不知道排序二分會不會有效。)

.. .
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。。應該就是最標準的掃描線。。因為圓有很多很好的性質,而且再結合有向面積代碼量要低於預想。。

首先開局出裝和上一種方法一樣。。直接進入主要部分。。。
。。枚舉每一個圓,計算所有其它圓和它的交點。。進行離散化。。這種環形離散化會產生一些問題。。參見 Master Spark 那支。。

然後大家就都會了吧。

… .. . .. .. .. . ..

。。完整代碼。。。。

/** ` 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);
}