【LOJ】 #2008. 「SCOI2015」小凸想跑步

题解

一道想法很简单的计算几何（由于我半平面交总是写不对，我理所当然的怀疑半平面交错了，事实上是我直线建错了）

首先我们对于两个凸包上的点设为\((x_0,y_0)\)和\((x_1,y_1)\)（逆时针）

设这个点为(x,y)我们用叉积求一下面积

可以得到

\((x_0 - x)(y_1 - y) - (x_1 - x)(y_0 - y)\)

\(x_0 y_1 - x_1 y_0 + (y_0 - y_1)x + (x_1 - x_0)y\)

然后我们可以对于每个小三角形都求一个这样的式子，我们的约束就是(x,y),P_0 P_1的三角形面积最小

最后会得到一个

\(Ax + By + C <= 0\)的式子

然后我们就可以建出直线来半平面交了，把x和C移到一边，讨论一下A是0或者B是0，然后讨论一下B的正负

半平面交求出来的凸包面积除以原凸包面积就是答案了

代码

#include <bits/stdc++.h>
#define MAXN 100005
//#define ivorysi
#define enter putchar('\n')
#define space putchar(' ')
#define fi first
#define se second
#define pb push_back
#define mp make_pair
#define eps 1e-8
#define pii pair<int,int>
using namespace std;
typedef long long int64;
typedef double db;
template<class T>
void read(T &res) {
    res = 0;char c = getchar();T f = 1;
    while(c < '0' || c > '9') {
	if(c == '-') f = -1;
	c = getchar();
    }
    while(c >= '0' && c <= '9') {
	res = res * 10 + c - '0';
	c = getchar();
    }
    res *= f;
}
template<class T>
void out(T x) {
    if(x < 0) {putchar('-');x = -x;}
    if(x >= 10) {
	out(x / 10);
    }
    putchar('0' + x % 10);
}
bool dcmp(db a,db b) {
    return fabs(a - b) < eps;
}
bool Gter(db a,db b) {
    return a > b + eps;
}
int N,tot;
struct Point {
    db x,y;
    Point(){}
    Point(db _x,db _y) {
	x = _x;y = _y;
    }
    friend Point operator + (const Point &a,const Point &b) {
	return Point(a.x + b.x,a.y + b.y);
    }
    friend Point operator - (const Point &a,const Point &b) {
	return Point(a.x - b.x,a.y - b.y);
    }
    friend Point operator * (const Point &a,const db &d) {
	return Point(a.x * d,a.y * d);
    }
    friend Point operator / (const Point &a,const db &d) {
	return Point(a.x / d,a.y / d);
    }
    friend db operator * (const Point &a,const Point &b) {
	return a.x * b.y - a.y * b.x;
    }
    friend db dot(const Point &a,const Point &b) {
	return a.x * b.x + a.y * b.y;
    }
}P[MAXN];
struct Seg {
    Point a,b;
    db d;
    Seg(){}
    Seg(Point _a,Point _b) {
	a = _a;b = _b;d = atan2(b.y - a.y,b.x - a.x);
    }
    friend Point Cross_Point(const Seg &s,const Seg &t) {
	db S1 = (s.a - t.a) * (t.b - t.a);
	db S2 = (s.b - t.b) * (t.a - t.b);
	return s.a + (s.b - s.a) * (S1 / (S1 + S2));
    }
}S[MAXN * 4];
bool cmp(Seg s,Seg t) {
    return Gter(t.d,s.d);
}
Point p[MAXN * 4];
Seg q[MAXN];
int ql,qr;
void Process() {
    ql = 1,qr = 0;
    for(int i = 1 ; i <= tot ; ++i) {
	while(ql < qr) {
	    if(Gter(0.0,(S[i].b - S[i].a) * (p[qr - 1] - S[i].a))) --qr;
	    else break;
	}
	while(ql < qr) {
	    if(Gter(0.0,(S[i].b - S[i].a) * (p[ql] - S[i].a))) ++ql;
	    else break;
	}
	q[++qr] = S[i];
	if(ql < qr) {
	    if(dcmp(q[qr].d,q[qr - 1].d)) {
		if(Gter((q[qr].b - q[qr].a) * (q[qr - 1].a - q[qr].a),0.0)) --qr;
		else q[qr - 1] = q[qr],--qr;
	    }
	}
	if(ql < qr) p[qr - 1] = Cross_Point(q[qr],q[qr - 1]);
    }
    while(ql < qr) {
	if(Gter(0.0,(q[ql].b - q[ql].a) * (p[qr - 1] - q[ql].a))) --qr;
	else break;
    }
    p[qr] = Cross_Point(q[qr],q[ql]);
    if(qr - ql + 1 >= 3) {
	db res = 0.0,s = 0.0;
	p[qr + 1] = p[ql];
	for(int i = ql ; i <= qr ; ++i) {
	    res += p[i] * p[i + 1];
	}
	for(int i = 0 ; i < N ; ++i) {
	    s += P[i] * P[i + 1];
	}
	printf("%.4lf\n",res / s);
    }
    else puts("0.0000");
}
void Solve() {
    read(N);
    db x,y;
    for(int i = 0 ; i < N ; ++i) {
	scanf("%lf%lf",&x,&y);
	P[i] = Point(x,y);
    }
    P[N] = P[0];
    for(int i = 0 ; i < N ; ++i) {
	S[++tot] = Seg(P[i],P[i + 1]);
    }
    db C0 = P[0] * P[1],A0 = P[0].y - P[1].y,B0 = P[1].x - P[0].x;
    for(int i = 1 ; i < N ; ++i) {
	db A = A0 - (P[i].y - P[i + 1].y);
	db B = B0 - (P[i + 1].x - P[i].x);
	db C = C0 - P[i] * P[i + 1];
	if(dcmp(A,0.0) && dcmp(B,0.0) && Gter(C,0.0)) {puts("0.0000");return;}
	if(dcmp(A,0.0)) {
	    if(Gter(B,0.0)) S[++tot] = Seg(Point(10,-C / B),Point(-10,-C / B));
	    else S[++tot] = Seg(Point(-10,-C / B),Point(10,-C / B));
	}
	else if(dcmp(B,0.0)) {
	    if(Gter(A,0.0)) S[++tot] = Seg(Point(-C / A,-10),Point(-C / A,10));
	    else S[++tot] = Seg(Point(-C / A,10),Point(-C / A,-10));
	}
	else {
	    A = -A / B,C = -C / B;
	    if(Gter(B,0.0)) S[++tot] = Seg(Point(10,A * 10 + C),Point(-10,-A * 10 + C));
	    else S[++tot] = Seg(Point(-10,-A * 10 + C),Point(10,A * 10 + C));
	}
    }
    sort(S + 1,S + tot + 1,cmp);
    Process();
}
int main() {
#ifdef ivorysi
    freopen("f1.in","r",stdin);
#endif
    Solve();
    return 0;
}