凸包&半平面求交 - 洛谷 - P7397 雨水收集系统

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往期相关背景半平面求交 点击前往
凸包点击前往

题目大意

题目链接
https://www.luogu.com.cn/problem/P7397

城市中有很多高楼，使用矩形表示。

云层用凸多边形表示，云层会移动，在特定的时间段下雨，云层经过的地方就会接受到雨。

求高楼能够接受到雨的面积总和。

解析

C到C’与虚线围成的区域既为降水区域，灰色为可以接受雨水的面积。
只要求出降水区域，再逐各与多边形求交即可。

凸多边形求交点击前往

降水区域多边形求解步骤：

1. 先求出云层所在的凸包 点击前往
2. 利用向量和降雨起止时间点得到C, C’位置。
3. 添加虚线，并删除中间线条，具体步骤如下图。
   
   上图中按照步骤1，2，3，4找到4个标记点，合围区域内的边可以删除。
   注意有边与移动方向平行的情况，需要用投影来判断取哪个。

tips:题目输入为整数，计算中能用整数尽量用整数，减少浮点误差，提升精度。矩阵坐标不一定是左下右上表示

代码

#include<stdio.h>
#include<cmath>
#include <algorithm>
#include <vector>
#include <list>
#include <cstring>
#include <set>


using namespace std;
const double EPS = 1e-6;

const int N = 2e3 + 10;

namespace FloatSys {
	int cmp(double d) {
		if (abs(d) < EPS)return 0;
		if (d > 0)return 1;
		return -1;
	}

	class Point {
	public:
		double x, y;
		int id;

		Point() {}
		Point(double a, double b) :x(a), y(b) {}
		Point(const Point& p) :x(p.x), y(p.y), id(p.id) {}

		void in() {
			scanf("%lf %lf", &x, &y);
		}
		void out() {
			printf("%.3f %.3f\n", x, y);
		}

		double dis() {
			return sqrt(x * x + y * y);
		}

		double dis2() {
			return x * x + y * y;
		}

		Point operator -() const {
			return Point(-x, -y);
		}

		Point operator -(const Point& p) const {
			return Point(x - p.x, y - p.y);
		}

		Point operator +(const Point& p) const {
			return Point(x + p.x, y + p.y);
		}
		Point operator *(double d)const {
			return Point(x * d, y * d);
		}

		Point operator /(double d)const {
			return Point(x / d, y / d);
		}


		void operator -=(Point& p) {
			x -= p.x;
			y -= p.y;
		}

		void operator +=(Point& p) {
			x += p.x;
			y += p.y;
		}
		void operator *=(double d) {
			x *= d;
			y *= d;
		}

		void operator /=(double d) {
			this ->operator*= (1 / d);
		}

		bool operator<(const Point& a) const {
			return x < a.x || (abs(x - a.x) < EPS && y < a.y);
		}

		bool operator==(const Point& a) const {
			return abs(x - a.x) < EPS && abs(y - a.y) < EPS;
		}
	};

	// 向量操作

	double cross(const Point& a, const Point& b) {
		return a.x * b.y - a.y * b.x;
	}


	double cross(const Point& a, const Point& b, const Point& c) {
		return cross(b - a, c - a);
	}

	double dot(const Point& a, const Point& b) {
		return a.x * b.x + a.y * b.y;
	}

	class Line {
	public:
		Point front, tail;
		double ang;
		int u, v;
		Line() {}
		Line(const Point& a, const Point& b) :front(a), tail(b) {
			ang = atan2(front.y - tail.y, front.x - tail.x);
		}
		void initAng() {
			ang = atan2(front.y - tail.y, front.x - tail.x);
		}
	};

	int cmp(const Line& a, const Line& b) {
		//if (a.u == b.u && a.v == b.v)return 0;
		return cmp(a.ang - b.ang);

	}


	// 点在直线哪一边>0 左边，<0边
	double SideJudge(const Line& a, const Point& b) {
		//return cmp(cross(a.front - a.tail, b - a.tail));
		return cross(a.front - a.tail, b - a.tail);
	}


	int LineSort(const Line& a, const Line& b) {
		int c = cmp(a, b);
		if (c)return c < 0;
		return	cross(b.front - b.tail, a.front - b.tail) > 0;
	}

	/*
	点p 到 p+r 表示线段1
	点q 到 q+s 表示线段2
	线段1 上1点用 p' = p+t*r (0<=t<=1)
	线段2 上1点用 q' = q+u*s (0<=u<=1)
	让两式相等求交点 p+t*r = q+u*s
	两边都叉乘s
	(p+t*r)Xs = (q+u*s)Xs
	pXs + t*rXs = qXs
	t = (q-p)Xs/(rXs)
	同理，
	u = (p-q)Xr/(sXr) -> u = (q-p)Xr/(rXs)

	以下分4种情况：
	1. 共线，sXr==0 && (q-p)Xr==0, 计算 (q-p)在r上的投影在r长度上的占比t0，
	计算(q+s-p)在r上的投影在r长度上的占比t1，查看[t0, t1]是否与范围[0，1]有交集。
	如果t0>t1, 则比较[t1, t0]是否与范围[0，1]有交集。
	t0 = (q-p)*r/(r*r)
	t1 = (q+s-p)*r/(r*r) = t0 + s · r / (r · r)
	2. 平行sXr==0 && (q-p)Xr!=0
	3. 0<=u<=1 && 0<=t<=1 有交点
	4. 其他u, t不在0到范围内，没有交点。
	*/
	pair<double, double> intersection(const Point& q, const Point& s, const Point& p, const Point& r, bool& oneline) {
		// 计算 (q-p)Xr
		auto qpr = cross(q - p, r);
		auto qps = cross(q - p, s);

		auto rXs = cross(r, s);
		if (cmp(rXs) == 0) {
			oneline = true;
			return { -1, -1 }; // 平行或共线
		}
		// 求解t, u
		// t = (q-p)Xs/(rXs)
		auto t = qps / rXs;

		// u = (q-p)Xr/(rXs)
		auto u = qpr / rXs;

		return { u, t };
	}

	Point LineCross(const Line& a, const Line& b, bool& f) {
		Point dira = a.front - a.tail;
		Point dirb = b.front - b.tail;
		bool oneline = false;
		auto p = intersection(a.tail, dira, b.tail, dirb, oneline);
		if (oneline)f = false;
		return a.tail + dira * p.first;
	}


	class HalfPlane {
	public:
		vector<Line> lines;

		void addLine(const Line& a) {
			lines.push_back(a);
		}

		vector<Point> run() {
			sort(lines.begin(), lines.end(), LineSort);
			vector<int> q(lines.size() + 10);
			vector<Point> t(lines.size() + 10);

			int l = -1, r = 0;
			q[0] = 0;
			for (int i = 1; i < lines.size(); ++i) {
				if (cmp(lines[i], lines[i - 1]) == 0)continue;
				while (r - l > 1 && SideJudge(lines[i], t[r]) < 0)r--;
				while (r - l > 1 && SideJudge(lines[i], t[l + 2]) < 0)l++;
				q[++r] = i;
				bool f = true;
				t[r] = LineCross(lines[q[r]], lines[q[r - 1]], f);
			}
			while (r - l > 1 && SideJudge(lines[q[l + 1]], t[r]) < 0)r--;
			if (r - l > 1) {
				bool f = true;
				t[r + 1] = LineCross(lines[q[l + 1]], lines[q[r]], f);
				r++;
			}

			// 统计交点
			l++;
			vector<Point> ans(r - l);
			for (int i = 0; i < ans.size(); ++i) {
				ans[i] = t[i + l + 1];
			}

			return ans;
		}
	};
}

typedef long long lld;

namespace IntSys {

	class Point {
	public:
		int x, y;
		int id;

		Point() {}
		Point(int a, int b) :x(a), y(b) {}
		Point(const Point& p) :x(p.x), y(p.y), id(p.id) {}

		void in() {
			scanf("%d %d", &x, &y);
		}
		void out() {
			printf("%d %d\n", x, y);
		}

		double dis() {
			return sqrt(x * x + y * y);
		}

		int dis2() {
			return x * x + y * y;
		}

		Point operator -() const {
			return Point(-x, -y);
		}

		Point operator -(const Point& p) const {
			return Point(x - p.x, y - p.y);
		}

		Point operator +(const Point& p) const {
			return Point(x + p.x, y + p.y);
		}
		Point operator *(int d)const {
			return Point(x * d, y * d);
		}

		/*Point operator /(double d)const {
			return Point(x / d, y / d);
		}*/


		void operator -=(Point& p) {
			x -= p.x;
			y -= p.y;
		}

		void operator +=(Point& p) {
			x += p.x;
			y += p.y;
		}
		void operator *=(int d) {
			x *= d;
			y *= d;
		}

		/*void operator /=(double d) {
			this ->operator*= (1 / d);
		}*/

		bool operator<(const Point& a) const {
			return x < a.x || (x == a.x && y < a.y);
		}

		bool operator==(const Point& a) const {
			return x == a.x && y == a.y;
		}
	};

	// 向量操作
	lld cross(const Point& a, const Point& b) {
		return lld(a.x) * lld(b.y) - lld(a.y) * lld(b.x);
	}

	lld cross(const Point& a, const Point& b, const Point& c) {
		return cross(b - a, c - a);
	}

	lld dot(const Point& a, const Point& b) {
		return lld(a.x) * lld(b.x) + lld(a.y) * lld(b.y);
	}
}


namespace CONVEX {

	IntSys::Point P[N];
	IntSys::Point lowPoint;
	int ind[N];
	int st[N];
	int top;


	bool cmp(int i, int j) {
		lld m = IntSys::cross(lowPoint, P[i], P[j]);
		if (m != 0) {
			return m > 0;
		}
		return (lowPoint - P[i]).dis2() < (lowPoint - P[j]).dis2();
	}

	void samcp(int n) {
		for (int i = 0; i < n; ++i)ind[i] = i;
		lowPoint = P[0]; // 最低最左边的点
		for (int i = 0; i < n; ++i) {
			if (P[i].y < lowPoint.y || (P[i].y == lowPoint.y && P[i].x < lowPoint.x)) lowPoint = P[i];
		}
		if (n == 1) {
			exit(1);
		}
		else if (n == 2) {
			exit(2);
		}

		sort(ind, ind + n, cmp);
		/*
		puts("");
		for (int i = 0; i < n; ++i) {
			printf("%d %d %d\n", P[ind[i]].x, P[ind[i]].y, ind[i]);
		}
		puts("");
		 */

		top = 2;
		st[0] = ind[0];
		st[1] = ind[1];
		for (int i = 2; i < n; ++i) {
			while (top >= 2 && IntSys::cross(P[st[top - 2]], P[ind[i]], P[st[top - 1]]) >= 0) {
				top--;
			}
			st[top++] = ind[i];
		}
		st[top] = ind[0];
	}

	double getArcLength() {
		double ans = 0;

		for (int i = 0; i < top; ++i) {
			ans += (P[st[i]] - P[st[i + 1]]).dis();
			// printf("%d %d\n", P[st[i]].x, P[st[i]].y);
		}

		return ans;
	}
}

IntSys::Point tangs[N];

int getMinute(int hh, int mm) {
	return hh * 60 + mm;
}

int gcd(int a, int b) {
	if (b == 0)return a;
	return gcd(b, a % b);
}

// cmpOp: 叉乘比较-1:<0, 1>0
// disOp: 叉乘为0时比较距离更远还更近，
int findKey(IntSys::Point dir, int cmpOp, int disOp) {
	using namespace CONVEX;
	int k = 0;
	for (int i = 1; i < CONVEX::top; ++i) {
		IntSys::Point p = CONVEX::P[st[i]] - P[st[k]];
		lld crossv = IntSys::cross(dir, p);
		if (crossv == 0) {
			// 比较距离
			lld dis1 = dot(dir, P[st[k]]);
			lld dis2 = dot(dir, P[st[i]]);
			if (dis1 == dis2)continue;
			dis2 -= dis1;
			dis2 /= abs(dis2);
			if (dis2 == disOp) k = i;
			continue;
		}

		// 不相等
		crossv /= abs(crossv);
		if (crossv == cmpOp) k = i;
	}

	return k;
}

void  solve() {
	int T;
	int n, m;
	int v;

	FloatSys::Point dir;
	IntSys::Point sp, ep;

	scanf("%d", &T);

	while (T--) {
		scanf("%d", &n);
		for (int i = 0; i < n; ++i) {
			tangs[i].in();
			tangs[i+n].in();
			// 不一定是左下右上，需要调整一下
			if (tangs[i].x > tangs[i + n].x)swap(tangs[i].x, tangs[i + n].x);
			if (tangs[i].y > tangs[i + n].y)swap(tangs[i].y, tangs[i + n].y);
		}
		scanf("%d", &m);

		for (int i = 0; i < m; ++i) {
			CONVEX::P[i].in();
		}
		sp.in();
		ep.in();
		scanf("%d", &v);
		int hh, mm;
		scanf("%d:%d", &hh, &mm);
		int t1 = getMinute(hh, mm);
		scanf("%d:%d", &hh, &mm);
		int t2 = getMinute(hh, mm);
		int totalTime = t2 - t1;
		//printf("%d %d %d\n", hh, mm, totalTime);
		//continue;
		dir.x = ep.x - sp.x;
		dir.y = ep.y - sp.y;
		dir /= dir.dis();
		dir *= v * totalTime;

		// 查找凸包关键点
		IntSys::Point diri = ep - sp;
		auto g = gcd(abs(diri.x), abs(diri.y));
		diri.x /= g;
		diri.y /= g;

		CONVEX::samcp(m);

		int k1, k2, k3, k4;
		k1 = findKey(diri, -1, 1);//最右边最前面点
		k2 = findKey(diri, 1, 1);//最左边最前面点
		k3 = findKey(diri, 1, -1);//最左边最后面点
		k4 = findKey(diri, -1, -1);//最右边最后面点

		FloatSys::HalfPlane hp;
		vector<FloatSys::Line> lines;
		for (int i = 0; i < 4; ++i)lines.push_back(FloatSys::Line());// 前面占用4个给矩形
		
		//添加初始云层线条
		for (int i = k1; i != k2; i = (i + 1) % CONVEX::top) {
			int i2 = (i + 1) % CONVEX::top;
			if (CONVEX::P[CONVEX::st[i]] == CONVEX::P[CONVEX::st[i2]])continue;
			lines.push_back(FloatSys::Line(FloatSys::Point(CONVEX::P[CONVEX::st[i2]].x, CONVEX::P[CONVEX::st[i2]].y)+dir,
				FloatSys::Point( CONVEX::P[CONVEX::st[i]].x, CONVEX::P[CONVEX::st[i]].y )+dir));
		}

		// 添加中间线
		lines.push_back(FloatSys::Line(FloatSys::Point(CONVEX::P[CONVEX::st[k3]].x, CONVEX::P[CONVEX::st[k3]].y),
			FloatSys::Point(CONVEX::P[CONVEX::st[k2]].x, CONVEX::P[CONVEX::st[k2]].y)+dir));

		//添加初始云层线条
		for (int i = k3; i != k4; i = (i + 1) % CONVEX::top) {
			int i2 = (i + 1) % CONVEX::top;
			if (CONVEX::P[CONVEX::st[i]] == CONVEX::P[CONVEX::st[i2]])continue;
			lines.push_back(FloatSys::Line(FloatSys::Point(CONVEX::P[CONVEX::st[i2]].x, CONVEX::P[CONVEX::st[i2]].y),
				FloatSys::Point(CONVEX::P[CONVEX::st[i]].x, CONVEX::P[CONVEX::st[i]].y)));
		}

		// 添加中间线
		lines.push_back(FloatSys::Line(FloatSys::Point(CONVEX::P[CONVEX::st[k1]].x, CONVEX::P[CONVEX::st[k1]].y)+dir,
			FloatSys::Point(CONVEX::P[CONVEX::st[k4]].x, CONVEX::P[CONVEX::st[k4]].y)));
		
		double ans = 0;
		// 对每个大厦与云层进行求交
		for (int i = 0; i < n; ++i) {

			lines[0].front.x = tangs[i + n].x;
			lines[0].front.y = tangs[i].y;
			lines[0].tail.x = tangs[i].x;
			lines[0].tail.y = tangs[i].y;
			lines[0].initAng();

			lines[1].front.x = tangs[i + n].x;
			lines[1].front.y = tangs[i + n].y;
			lines[1].tail.x = tangs[i + n].x;
			lines[1].tail.y = tangs[i].y;
			lines[1].initAng();

			lines[2].front.x = tangs[i].x;
			lines[2].front.y = tangs[i + n].y;
			lines[2].tail.x = tangs[i + n].x;
			lines[2].tail.y = tangs[i + n].y;
			lines[2].initAng();

			lines[3].front.x = tangs[i].x;
			lines[3].front.y = tangs[i].y;
			lines[3].tail.x = tangs[i].x;
			lines[3].tail.y = tangs[i + n].y;
			lines[3].initAng();

			hp.lines = lines;
			/*printf("%d\n", hp.lines.size());

			for (auto l : hp.lines) {
				l.tail.out();
				l.front.out();
			}*/

			auto hpps = hp.run();
			//puts("lines end");
			//printf("hpps size: %d\n", hpps.size());
			/*hpps[0].out();
			hpps[1].out();*/
			for (int j = 2; j < hpps.size(); ++j) {
				//hpps[j].out();
				ans += FloatSys::cross(hpps[0], hpps[j - 1], hpps[j]);
			}

			//printf("# %d: \n", i);
		}

		printf("%.3f\n", ans/2);
	}

}


int main() {
	solve();
	return 0;

}

/*
2

2
0 0 10 10
20 20 30 10
4
-10 8 -5 8 -5 13 -10 13
15 0 25 0
1
15:30 16:05

2
0 0 10 10
20 20 30 10
4
-10 8 -5 8 -5 13 -10 13
-5 8 19 1
1
15:30 16:30


1

1
20 20 30 10
4
-10 8 -5 8 -5 13 -10 13
15 0 25 0
1
15:30 16:05


2

2
0 0 10 10
20 20 30 10
4
-10 -10 -5 -10 -5 -5 -10 -5
15 0 25 0
1
15:30 16:05

*/

---

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