2019 牛客多校第一场 F Random Point in Triangle

题目链接：https://ac.nowcoder.com/acm/contest/881/F

题目大意

　　给定二维平面上 3 个整数表示的点 A，B，C，在三角形 ABC 内随机选一点 P，求期望$E = max(S_{PAB}, S_{PAC}, S_{PBC})$。输出 36 * E。

分析

　　先说结论，答案是$22S_{ABC}$，证明如下：

　　不妨设 A 为 (0, 0)，B 为 (1, 0), C 为 (a, b)，这是因为对于任意一个三角形，总可以把 A 点移动到原点，然后旋转使 AB 与 x 轴重合，然后缩放使 AB = 1。

　　设 P 为 (x, y)。

　　设重心为 G，各点对应边中点分别为 A', B', C'。

　　由于$S_{PAB}, S_{PAC}, S_{PBC}$地位是相等的，因此下面只讨论$E = S_{PAB}$的情况，也就是 P 落在四边形 CB'GA' 内部的情况。

　　简图如下：

 

　　以下列出一些点的坐标和一些直线的函数方程：

• G：$(\frac{a + 1}{3}, \frac{b}{3})$
• B'：$(\frac{a}{2}, \frac{b}{2})$
• A'：$(\frac{a + 1}{2}, \frac{b}{2})$
• B'A'：$x = \frac{b}{2}$
• CA'：$y = \frac{b}{a - 1}(x - 1)$
• CB'：$y = \frac{b}{a}x$
• GB'：$y = \frac{b}{a - 2}(x - 1)$
• GA'：$y = \frac{b}{a + 1}x$
• $S_{PAB}$：$\frac{y}{2}$
• $S_{CB'GA'}$：$\frac{b}{6}$
• $S_{CB'A'}$：$\frac{b}{8}$
• $S_{GB'A'}$：$\frac{b}{24}$

　　于是$E = \frac{1}{S_{CB'GA'}} ((\int_{G_y}^{A'B'} dy \int_{GB'}^{GA'} S_{PAB} dx) + (\int_{A'B'}^{C_y} dy \int_{CB'}^{CA'} S_{PAB} dx))$

　　积出来得$E = \frac{11b}{36} = \frac{11}{18}S_{ABC}$

　　所以$36E = 22S_{ABC}$

代码如下

  1 #include <bits/stdc++.h>
  2 using namespace std;
  3  
  4 #define INIT() ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
  5 #define Rep(i,n) for (int i = 0; i < (n); ++i)
  6 #define For(i,s,t) for (int i = (s); i <= (t); ++i)
  7 #define rFor(i,t,s) for (int i = (t); i >= (s); --i)
  8 #define ForLL(i, s, t) for (LL i = LL(s); i <= LL(t); ++i)
  9 #define rForLL(i, t, s) for (LL i = LL(t); i >= LL(s); --i)
 10 #define foreach(i,c) for (__typeof(c.begin()) i = c.begin(); i != c.end(); ++i)
 11 #define rforeach(i,c) for (__typeof(c.rbegin()) i = c.rbegin(); i != c.rend(); ++i)
 12  
 13 #define pr(x) cout << #x << " = " << x << "  "
 14 #define prln(x) cout << #x << " = " << x << endl
 15  
 16 #define LOWBIT(x) ((x)&(-x))
 17  
 18 #define ALL(x) x.begin(),x.end()
 19 #define INS(x) inserter(x,x.begin())
 20 #define UNIQUE(x) x.erase(unique(x.begin(), x.end()), x.end())
 21 #define REMOVE(x, c) x.erase(remove(x.begin(), x.end(), c), x.end()); // ?? x ?????? c 
 22 #define TOLOWER(x) transform(x.begin(), x.end(), x.begin(),::tolower);
 23 #define TOUPPER(x) transform(x.begin(), x.end(), x.begin(),::toupper);
 24  
 25 #define ms0(a) memset(a,0,sizeof(a))
 26 #define msI(a) memset(a,inf,sizeof(a))
 27 #define msM(a) memset(a,-1,sizeof(a))
 28 
 29 #define MP make_pair
 30 #define PB push_back
 31 #define ft first
 32 #define sd second
 33  
 34 template<typename T1, typename T2>
 35 istream &operator>>(istream &in, pair<T1, T2> &p) {
 36     in >> p.first >> p.second;
 37     return in;
 38 }
 39  
 40 template<typename T>
 41 istream &operator>>(istream &in, vector<T> &v) {
 42     for (auto &x: v)
 43         in >> x;
 44     return in;
 45 }
 46 
 47 template<typename T>
 48 ostream &operator<<(ostream &out, vector<T> &v) {
 49     Rep(i, v.size()) out << v[i] << " \n"[i == v.size()];
 50     return out;
 51 }
 52 
 53 template<typename T1, typename T2>
 54 ostream &operator<<(ostream &out, const std::pair<T1, T2> &p) {
 55     out << "[" << p.first << ", " << p.second << "]" << "\n";
 56     return out;
 57 }
 58 
 59 inline int gc(){
 60     static const int BUF = 1e7;
 61     static char buf[BUF], *bg = buf + BUF, *ed = bg;
 62     
 63     if(bg == ed) fread(bg = buf, 1, BUF, stdin);
 64     return *bg++;
 65 } 
 66 
 67 inline int ri(){
 68     int x = 0, f = 1, c = gc();
 69     for(; c<48||c>57; f = c=='-'?-1:f, c=gc());
 70     for(; c>47&&c<58; x = x*10 + c - 48, c=gc());
 71     return x*f;
 72 }
 73 
 74 template<class T>
 75 inline string toString(T x) {
 76     ostringstream sout;
 77     sout << x;
 78     return sout.str();
 79 }
 80 
 81 inline int toInt(string s) {
 82     int v;
 83     istringstream sin(s);
 84     sin >> v;
 85     return v;
 86 }
 87 
 88 //min <= aim <= max
 89 template<typename T>
 90 inline bool BETWEEN(const T aim, const T min, const T max) {
 91     return min <= aim && aim <= max;
 92 }
 93  
 94 typedef long long LL;
 95 typedef unsigned long long uLL;
 96 typedef pair< double, double > PDD;
 97 typedef pair< int, int > PII;
 98 typedef pair< int, PII > PIPII;
 99 typedef pair< string, int > PSI;
100 typedef pair< int, PSI > PIPSI;
101 typedef set< int > SI;
102 typedef set< PII > SPII;
103 typedef vector< int > VI;
104 typedef vector< double > VD;
105 typedef vector< VI > VVI;
106 typedef vector< SI > VSI;
107 typedef vector< PII > VPII;
108 typedef map< int, int > MII;
109 typedef map< int, string > MIS;
110 typedef map< int, PII > MIPII;
111 typedef map< PII, int > MPIII;
112 typedef map< string, int > MSI;
113 typedef map< string, string > MSS;
114 typedef map< PII, string > MPIIS;
115 typedef map< PII, PII > MPIIPII;
116 typedef multimap< int, int > MMII;
117 typedef multimap< string, int > MMSI;
118 //typedef unordered_map< int, int > uMII;
119 typedef pair< LL, LL > PLL;
120 typedef vector< LL > VL;
121 typedef vector< VL > VVL;
122 typedef priority_queue< int > PQIMax;
123 typedef priority_queue< int, VI, greater< int > > PQIMin;
124 const double EPS = 1e-8;
125 const LL inf = 0x7fffffff;
126 const LL infLL = 0x7fffffffffffffffLL;
127 const LL mod = 1e9 + 7;
128 const int maxN = 2e5 + 7;
129 const LL ONE = 1;
130 const LL evenBits = 0xaaaaaaaaaaaaaaaa;
131 const LL oddBits = 0x5555555555555555;
132 
133 template<typename T>
134 struct Point{
135     T X,Y;
136     Point< T >(T x = 0,T y = 0) : X(x), Y(y) {}
137         
138     inline Point<T> operator* (const T& k) { return Point<T>(k*X, k*Y); }
139     inline Point<T> operator/ (const T& k) { return Point<T>(X/k, Y/k); }
140     inline Point<T> operator+ (const Point<T>& x) { return Point<T>(x.X+X,x.Y+Y); }
141     inline Point<T> operator- (const Point<T>& x) { return Point<T>(x.X-X,x.Y-Y); }
142     
143     T operator^(const Point<T> &x) const { return X*x.Y - Y*x.X; }
144     T operator*(const Point<T> &x) const { return X*x.X + Y*x.Y; }
145 };
146 
147 template<typename T>
148 istream &operator>> (istream &in, Point<T> &x) {
149     in >> x.X >> x.Y;
150     return in;
151 }
152 
153 Point< LL > p[3];
154 
155 int main(){
156     //freopen("MyOutput.txt","w",stdout);
157     //freopen("input.txt","r",stdin);
158     //INIT();
159     while(cin >> p[0] >> p[1] >> p[2]) cout << 11 * abs((p[0] - p[1]) ^ (p[0] - p[2])) << endl;
160     return 0;
161 }

View Code

 

 

转载于:https://www.cnblogs.com/zaq19970105/p/11239824.html