洛谷

https://www.luogu.org/problemnew/show/P1663

给定山的性状，求一个最低点可以看见所有的地方。

就是半平面交。

粘贴全家福：

```cpp #include using namespace std; typedef long long ll;

//不要输出-0.0之类的数

const double eps=1e-8;
const double inf=1e9;
const double pi=acos(-1.0); //小数点后15位精度，和atan2相同

//判断浮点数的符号
inline int cmp(double x) {
return (fabs(x)<eps)?0:((x>0.0)?1:-1);
}

inline double sqr(double x) {
return x*x;
}

struct Point {
double x,y;
Point() {};
Point(const double x,const double 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 &p,const double k) {
    return Point(p.x*k,p.y*k);
}
friend Point operator*(const double k,const Point &p) {
    return Point(p.x*k,p.y*k);
}
friend Point operator/(const Point &p,const double k) {
    return Point(p.x/k,p.y/k);
}
inline double angle() {
    //返回向量的倾斜角，[-pi, pi]
    return atan2(y,x);
}

};

double det(const Point &a,const Point &b) {
return a.xb.y-a.yb.x;
}
double dot(const Point &a,const Point &b) {
return a.xb.x+a.yb.y;
}

//半平面交，O(nlogn)
vector halfplane_intersection(vector v) {
sort(v.begin(),v.end(),compare);
deque q;
deque ans;
q.push_back(v[0]);

int vs=v.size();
for(int i=1;i<vs;++i){
    if(cmp((v[i].second-v[i].first).angle()-(v[i-1].second-v[i-1].first).angle())==0)
        continue;
    while(!ans.empty()&&!satisfy(ans.back(),v[i])){
        ans.pop_back();
        q.pop_back();
    }
    while(!ans.empty()&&!satisfy(ans.front(),v[i])){
        ans.pop_front();
        q.pop_front();
    }
    ans.push_back(intersect_point(q.back(),v[i]));
    q.push_back(v[i]);
}
while(!ans.empty()&&!satisfy(ans.back(),q.front())){
    ans.pop_back();
    q.pop_back();
}
while(!ans.empty()&&!satisfy(ans.front(),q.back())){
    ans.pop_front();
    q.pop_front();
}
ans.push_back(intersect_point(q.back(),q.front()));
return vector<Point>(ans.begin(),ans.end());

}

void solve() {
int n;
scanf("%d",&n);
double x[5005],y[5005];
for(int i=1;i<=n;i++){
scanf("%lf%lf",&x[i],&y[i]);
}
vector hp;
hp.push_back(Halfplane(Point(x[1],inf),Point(x[1],y[1])));
hp.push_back(Halfplane(Point(x[n],y[n]),Point(x[n],inf)));
hp.push_back(Halfplane(Point(x[n],inf),Point(x[1],inf)));
for(int i=2;i<=n;i++){
hp.push_back(Halfplane(Point(x[i-1],y[i-1]),Point(x[i],y[i])));
}

/*for(int i=0;i<(int)hp.size();i++){
    printf("Point1 %d: (%.4f, %4f)
",i+1,hp[i].first.x,hp[i].first.y);
    printf("Point2 %d: (%.4f, %4f)
",i+1,hp[i].second.x,hp[i].second.y);

    puts("");
}*/

double ans=inf;
vector<Point> hpi=halfplane_intersection(hp);
int hs=hpi.size();
for(int i=0;i<hs;i++){
    //printf("Point %d: (%.4f, %4f)
",i+1,hpi[i].x,hpi[i].y);
    ans=min(ans,hpi[i].y);
}
printf("%8f
",ans);

}

int main() {

ifdef Yinku

freopen("Yinku.in","r",stdin);

endif // Yinku

solve();
return 0;

}

<details>