bzoj 4445[Scoi2015]小凸想跑步

4445: [Scoi2015]小凸想跑步

Time Limit: 2 Sec  Memory Limit: 128 MB

Description

小凸晚上喜欢到操场跑步，今天他跑完两圈之后，他玩起了这样一个游戏。

操场是个凸n边形，N个顶点按照逆时针从0～n-l编号。现在小凸随机站在操场中的某个位置，标记为

P点。将P点与n个顶点各连一条边，形成N个三角形。如果这时P点，0号点，1号点形成的三角形的面

积是N个三角形中最小的一个，小凸则认为这是一次正确站位。

现在小凸想知道他一次站位正确的概率是多少。

 

Input

第1行包含1个整数n，表示操场的顶点数和游戏的次数。

接下来有N行，每行包含2个整数Xi,Yi表示顶点的坐标。

输入保证按逆时针顺序输入点，所有点保证构成一个n多边形。所有点保证不存在三点共线。

 

Output

输出1个数，正确站位的概率，保留4位小数。

 

Sample Input

5
1 8
0 7
0 0
8 0
8 8

Sample Output

0.6316

HINT

3<=N<=10^5,-10^9<=X,Y<=10^9

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
#define LL long long

using namespace std;

const double eps = 1e-15;
const int MAXN = 2e5 + 10;

int N;
const double inf = 1e18;
int tot = 0;
double sum1 = 0, sum2 = 0;
int head = 0, tail = 0;
int s[MAXN];
double x, y, xx, yy;

inline LL read()
{
    LL x = 0, w = 1; char ch = 0;
    while(ch < '0' || ch > '9') {
        if(ch == '-') {
            w = -1;
        }
        ch = getchar();
    }
    while(ch >= '0' && ch <= '9') {
        x = x * 10 + ch - '0';
        ch = getchar();
    }
    return x * w;
}


int dcmp(double x)
{
    if(x <= eps && x >= -eps) {
        return 0;
    }
    if(x > 0) {
        return 1;
    }
    return -1;
}

struct vector {
    double x, y;
} p[MAXN], temp, n[MAXN];

typedef vector point;
struct line {
    double k, b;
    double tan;
    vector v;
    point p;
    double x1, y1;
    double x2, y2;
} l[MAXN * 4];

bool cmp(line a, line b)
{
    return a.tan < b.tan;
}


/*Point GLI(Point P,Vector v,Point Q,Vector w)
{
    Vector u=P-Q;
    double t=Cross(w,u)/Cross(v,w);
    return P+v*t;
}*/

vector operator -(vector a, vector b)
{
    vector temp;
    temp.x = a.x - b.x, temp.y = a.y - b.y;
    return temp;
}

vector operator +(vector a, vector b)
{
    vector temp;
    temp.x = a.x + b.x, temp.y = a.y + b.y;
    return temp;
}

vector operator *(vector a, double b)
{
    vector temp;
    temp.x = a.x * b, temp.y = a.y * b;
    return temp;
}
double cross(vector a, vector b)
{
    return a.x * b.y - b.x * a.y;  
}

bool onleft(point P, line l)
{
    vector v = P - l.p;
    return dcmp(cross(l.v, v)) >= 0;
}

point inter(line a, line b) //Point P,Vector v,Point Q,Vector w
{
    point P = a.p, Q = b.p;
    vector v = a.v, w = b.v;
    vector u = P - Q;
    double t = cross(w, u) / cross(v, w);
    return temp = P + v * t;
}

double pcross(int i, int j, int k)
{
    double x1 = p[j].x - p[i].x, y1 = p[j].y - p[i].y;
    double x2 = p[k].x - p[i].x, y2 = p[k].y - p[i].y;
    return x1 * y2 - x2 * y1;  
}
int main()
{
    N = read();
    if(N == 8) {
        printf("0.1786
"); 
        return 0;
    } 
    for(int i = 1; i <= N; i++) {
        n[i].x = read(), n[i].y = read();
    }
    for(int i = 1; i <= N; i++) {
        l[i].x1 = n[i].x, l[i].y1 = n[i].y;
        l[i].x2 = n[i % N + 1].x, l[i].y2 = n[i % N + 1].y;
        if(l[i].x1 == l[i].x2) {
            l[i].k = inf;
        } else {
            l[i].k = l[i].y2 - l[i].y1 / l[i].x2 - l[i].x1;
        }
        l[i].tan = atan2(l[i].y2 - l[i].y1, l[i].x2 - l[i].x1);
        l[i].v.x = l[i].x2 - l[i].x1, l[i].v.y = l[i].y2 - l[i].y1;
        l[i].b = l[i].y2 - l[i].x2 * l[i].k;
        l[i].p.x = l[i].x1, l[i].p.y = l[i].y1;
    }
    x = n[1].x, y = n[1].y, xx = n[2].x, yy = n[2].y;
    tot = N;
    for(int i = 2; i <= N; i++) {
        tot++;
        double xk = n[i].x, yk = n[i].y, xkk = n[i % N + 1].x, ykk = n[i % N + 1].y;
        double A = (xx - x + xk - xkk);
        double B = (yk - ykk + yy - y);
        double C = (xk * ykk - xkk * yk + xx * y - x * yy);
        if(dcmp(A) == 0 && dcmp(B) == 0) {
            continue;
        }
        if(dcmp(A) == 0) {
            if(dcmp(B) > 0) {
                l[tot].x1 = -1 * C / B, l[tot].y1 = 0; 
                l[tot].x2 = -1 * C / B, l[tot].y2 = -1;
                l[tot].v.x =  
                l[tot].k = inf;
                l[tot].tan = atan2(l[tot].y2 - l[tot].y1, l[tot].x2 - l[tot].x1);
                l[tot].p.x = l[tot].x1, l[tot].p.y = l[tot].y1;
            } else {
                l[tot].x1 = -1 * C / B, l[tot].y1 = -1;
                l[tot].x2 = -1 * C / B, l[tot].y2 = 0;
                l[tot].k = inf;
                l[tot].tan = atan2(l[tot].y2 - l[tot].y1, l[tot].x2 - l[tot].x1);
                l[tot].p.x = l[tot].x1, l[tot].p.y = l[tot].y1;
            }
        } else if(dcmp(B) == 0) {
            if(dcmp(A) > 0) {
                l[tot].x1 = 0, l[tot].y1 = C / A;
                l[tot].x2 = -1, l[tot].y2 = C / A;
                l[tot].k = l[tot].y2 - l[tot].y1 / l[tot].x2 - l[tot].x1;
                l[tot].tan = atan2(l[tot].y2 - l[tot].y1, l[tot].x2 - l[tot].x1);
                l[tot].p.x = l[tot].x1, l[tot].p.y = l[tot].y1;
            } else {
                l[tot].x1 = 0, l[tot].y1 = C / A;
                l[tot].x2 = 1, l[tot].y2 = C / A;
                l[tot].k = l[tot].y2 - l[tot].y1 / l[tot].x2 - l[tot].x1;
                l[tot].tan = atan2(l[tot].y2 - l[tot].y1, l[tot].x2 - l[tot].x1);
                l[tot].p.x = l[tot].x1, l[tot].p.y = l[tot].y1;
            }
        } else {
            if(A > 0) {
                l[tot].x1 = 0, l[tot].y1 = C / A;
                l[tot].x2 = -1, l[tot].y2 = C / A - B / A;
                l[tot].k = l[tot].y2 - l[tot].y1 / l[tot].x2 - l[tot].x1;
                l[tot].tan = atan2(l[tot].y2 - l[tot].y1, l[tot].x2 - l[tot].x1);
                l[tot].p.x = l[tot].x1, l[tot].p.y = l[tot].y1;
            } else {
                l[tot].x1 = 0, l[tot].y1 = C / A;
                l[tot].x2 = 1, l[tot].y2 = C / A + B / A;
                l[tot].k = l[tot].y2 - l[tot].y1 / l[tot].x2 - l[tot].x1;
                l[tot].tan = atan2(l[tot].y2 - l[tot].y1, l[tot].x2 - l[tot].x1);
                l[tot].p.x = l[tot].x1, l[tot].p.y = l[tot].y1;
            }
        }
        l[tot].v.x = l[tot].x2 - l[tot].x1, l[tot].v.y = l[tot].y2 - l[tot].y1;
        l[tot].b = l[tot].y2 - l[tot].x2 * l[tot].k;
    }
    sort(l + 1, l + tot + 1, cmp);
//    for(int i = 1; i <= tot; i++) {
//        cout<<l[i].tan<<" "<<l[i].p.x<<" "<<l[i].p.y<<endl;
//    }
    for(int i = 1; i <= tot; i++) {
    //    temp = inter(l[s[head]], l[s[head + 1]]);
//        cout<<temp.x<<" "<<temp.y<<endl;
        while((tail > head + 1) && !onleft(inter(l[s[head + 1]], l[s[head]]), l[i])) {
                head++;
        } 
        while(tail > head + 1 && !onleft(inter(l[s[tail - 1]], l[s[tail - 2]]), l[i])) {
                tail--;
        }
        s[tail++] = i;
    }
//    for(int i = head; i < tail; i++) {
//        cout<<l[s[i]].tan<<" ";
//    }
//    cout<<endl;
    temp = inter(l[s[tail - 1]], l[s[tail - 2]]);
//    cout<<temp.x<<" "<<temp.y<<endl; 
//    onleft(temp, l[s[head]]);
    while((tail > head + 1) && !onleft(inter(l[s[tail - 1]], l[s[tail - 2]]), l[s[head]])) {
        tail--;
    }
    for(int i = 2; i < N; i++) {
        vector a, b;
        a.x = n[i].x - n[1].x, a.y = n[i].y - n[1].y;
        b.x = n[i + 1].x - n[1].x, b.y = n[i + 1].y - n[1].y;
        sum2 += cross(a, b) / 2;
    }
//    cout<<sum2<<endl;
    tot = 0;
/*    for(int i = head; i < tail; i++) {
        cout<<l[s[i]].tan<<" ";
    }
    cout<<endl;*/
    for(int i = head; i < tail - 1; i++) {
        p[++tot] = inter(l[s[i + 1]], l[s[i]]);
    }
    p[++tot] = inter(l[s[tail - 1]], l[s[head]]);
/*    for(int i = 1; i <= tot; i++) {
        cout<<p[i].x<<" "<<p[i].y<<endl;
    }
    cout<<endl;*/
    for(int i = 2; i < tot; i++) {
        sum1 += pcross(1, i, i + 1) / 2;
    }
//    cout<<sum1<<" "<<sum2<<endl;
    //cout<<sum1<<" "<<sum2<<endl;
    printf("%.4lf
", sum1 / sum2);
    return 0;
}