洛谷 P4196 [CQOI2006]凸多边形 (半平面交)

题目链接：P4196 [CQOI2006]凸多边形

题意

给定 \(n\) 个凸多边形，求它们相交的面积。

思路

半平面交

半平面交的模板题。

代码

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef double db;
const db eps = 1e-10;
const db pi = acos(-1.0);
const ll inf = 0x3f3f3f3f3f3f3f3f;
const ll maxn = 1e3 + 10;

inline int dcmp(db x) {
    if(fabs(x) < eps) return 0;
    return x > 0? 1: -1;
}

class Point {
public:
    double x, y;
    Point(double x = 0, double y = 0) : x(x), y(y) {}
    inline void input() {
        scanf("%lf%lf", &x, &y);
    }
    bool operator<(const Point &a) const {
        return (!dcmp(x - a.x))? dcmp(y - a.y) < 0: x < a.x;
    }
    bool operator==(const Point &a) const {
        return dcmp(x - a.x) == 0 && dcmp(y - a.y) == 0;
    }
    db dis2(const Point a) {
        return pow(x - a.x, 2) + pow(y - a.y, 2);
    }
    db dis(const Point a) {
        return sqrt(dis2(a));
    }

    db dis2() {
        return x * x + y * y;
    }
    db dis() {
        return sqrt(dis2());
    }
    Point operator+(const Point a) {
        return Point(x + a.x, y + a.y);
    }
    Point operator-(const Point a) {
        return Point(x - a.x, y - a.y);
    }
    Point operator*(double p) {
        return Point(x * p, y * p);
    }
    Point operator/(double p) {
        return Point(x / p, y / p);
    }
    db dot(const Point a) {
        return x * a.x + y * a.y;
    }
    db cross(const Point a) {
        return x * a.y - y * a.x;
    }
    db ang(Point a) {
        return acos((a.dis() * dis()) / dot(a));
    }
};
typedef Point Vector;

Point p[maxn], ip[maxn];

class Line {
public:
    Point s, e;
    db angle;
    Line() {}
    Line(Point s, Point e) : s(s), e(e) {}
    inline void input() {
        s.input();e.input();
    }
    bool operator<(const Line &a) const {
        Line l = a;
        if(dcmp(angle - l.angle) == 0) {
            return l.toLeftTest(s) == 1;
        }
        return angle < l.angle;
    }
    void get_angle() {
        angle = atan2(e.y - s.y, e.x - s.x);
    }
    int toLeftTest(Point p) {
        if((e - s).cross(p - s) > 0) return 1;
        else if((e - s).cross(p - s) < 0) return -1;
        return 0;
    }
    int linecrossline(Line l) {
        if(dcmp((e - s).cross(l.e - l.s)) == 0) {
            if(dcmp((l.s - e).cross(l.e - s)) == 0) {
                return 0;
            }
            return 1;
        }
        return 2;
    }
    Point crosspoint(Line l) {
		db a1 = (l.e - l.s).cross(s - l.s);
		db a2 = (l.e - l.s).cross(e - l.s);
        db x = (s.x * a2 - e.x * a1) / (a2 - a1);
        db y = (s.y * a2 - e.y * a1) / (a2 - a1);
        if(dcmp(x) == 0) x = 0;
        if(dcmp(y) == 0) y = 0;
		return Point(x, y);
	}
};

Line l[maxn], q[maxn];

db half_plane(int cnt) {
    sort(l + 1, l + 1 + cnt);
    // for(int i = 1; i <= cnt; ++i) {
    //     cout << l[i].s.x << " " << l[i].s.y << " ";
    //     cout << l[i].e.x << " " << l[i].e.y << endl;
    // }
    int tmp = 1;
    for(int i = 2; i <= cnt; ++i) {
        if(dcmp(l[i].angle - l[tmp].angle) == 1) l[++tmp] = l[i];
    }
    cnt = tmp;
    int head = 1, tail = 2;
    q[1] = l[1], q[2] = l[2];
    for(int i = 3; i <= cnt; ++i) {
        while(head < tail && l[i].toLeftTest(q[tail].crosspoint(q[tail - 1])) == -1) {
            --tail;
        }
        while(head < tail && l[i].toLeftTest(q[head].crosspoint(q[head + 1])) == -1) {
            ++head;
        }
        q[++tail] = l[i];
    }

    while(head < tail && q[head].toLeftTest(q[tail].crosspoint(q[tail - 1])) == -1) {
        --tail;
    }
    while(head < tail && q[tail].toLeftTest(q[head].crosspoint(q[head + 1])) == -1) {
        ++head;
    }

    if(tail - head + 1 <= 2) {
        return 0.0;
    }

    tmp = 0;
    for(int i = head; i < tail; ++i) {
        ip[++tmp] = q[i].crosspoint(q[i + 1]);
    }
    ip[++tmp] = q[head].crosspoint(q[tail]);
    db ans = 0;
    for(int i = 3; i <= tmp; ++i) {
        ans += (ip[i - 1] - ip[1]).cross(ip[i] - ip[1]);
    }
    return ans * 0.5;
}

int main() {
    int n;
    scanf("%d", &n);
    int cnt = 0;
    for(int i = 1; i <= n; ++i) {
        int m;
        scanf("%d", &m);
        for(int i = 0; i < m; ++i) {
            p[i].input();
            if(i) {
                l[++cnt].e = p[i];
                l[cnt].s = p[i - 1];
                l[cnt].get_angle();
            }
        }
        l[++cnt].e = p[0];
        l[cnt].s = p[m - 1];
        l[cnt].get_angle();
    }
    printf("%.3lf\n", half_plane(cnt));
    return 0;
}