1396 - Most Distant Point from the Sea

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题意：
按顺序给出一小岛（多边形）的点
求岛上某点离海最远的距离
解法：
不断的收缩多边形（求半平面交）
直到无限小
二分收缩的距离即可
如图
//大白p263

#include <cmath>

#include <cstdio>

#include <cstring>

#include <string>

#include <queue>

#include <functional>

#include <set>

#include <iostream>

#include <vector>

#include <algorithm>

using namespace std;

const double eps=1e-7;//精度

const int INF=0x3f3f3f3f;

const double PI=acos(-1.0);

int dcmp(double x){//判断double等于0或。。。

	if(fabs(x)<eps)return 0;else return x<0?-1:1;

}

struct Point{

	double x,y;

	Point(double x=0,double y=0):x(x),y(y){}

};

typedef Point Vector;

typedef vector<Point> Polygon;

Vector operator+(Vector a,Vector b){return Vector(a.x+b.x,a.y+b.y);}//向量+向量=向量

Vector operator-(Point a,Point b){return Vector(a.x-b.x,a.y-b.y);}//点-点=向量

Vector operator*(Vector a,double p){return Vector(a.x*p,a.y*p);}//向量*实数=向量

Vector operator/(Vector a,double p){return Vector(a.x/p,a.y/p);}//向量/实数=向量

bool operator<( const Point& A,const Point& B ){return dcmp(A.x-B.x)<0||(dcmp(A.x-B.x)==0&&dcmp(A.y-B.y)<0);}

bool operator==(const Point&a,const Point&b){return dcmp(a.x-b.x)==0&&dcmp(a.y-b.y)==0;}

bool operator!=(const Point&a,const Point&b){return a==b?false:true;}

struct Segment{

	Point a,b;

	Segment(){}

	Segment(Point _a,Point _b){a=_a,b=_b;}

	bool friend operator<(const Segment& p,const Segment& q){return p.a<q.a||(p.a==q.a&&p.b<q.b);}

	bool friend operator==(const Segment& p,const Segment& q){return (p.a==q.a&&p.b==q.b)||(p.a==q.b&&p.b==q.a);}

};

struct Circle{

	Point c;

	double r;

	Circle(){}

	Circle(Point _c, double _r):c(_c),r(_r) {}

	Point point(double a)const{return Point(c.x+cos(a)*r,c.y+sin(a)*r);}

	bool friend operator<(const Circle& a,const Circle& b){return a.r<b.r;}

};

struct Line{

	Point p;

	Vector v;

	double ang;

	Line() {}

	Line(const Point &_p, const Vector &_v):p(_p),v(_v){ang = atan2(v.y, v.x);}

	bool operator<(const Line &L)const{return  ang < L.ang;}

};

double Dot(Vector a,Vector b){return a.x*b.x+a.y*b.y;}//|a|*|b|*cosθ 点积

double Length(Vector a){return sqrt(Dot(a,a));}//|a| 向量长度

double Angle(Vector a,Vector b){return acos(Dot(a,b)/Length(a)/Length(b));}//向量夹角θ

double Cross(Vector a,Vector b){return a.x*b.y-a.y*b.x;}//叉积 向量围成的平行四边形的面积

double Area2(Point a,Point b,Point c){return Cross(b-a,c-a);}//同上 参数为三个点

double DegreeToRadius(double deg){return deg/180*PI;}

double GetRerotateAngle(Vector a,Vector b){//向量a顺时针旋转theta度得到向量b的方向

	double tempa=Angle(a,Vector(1,0));

	if(a.y<0) tempa=2*PI-tempa;

	double tempb=Angle(b,Vector(1,0));

	if(b.y<0) tempb=2*PI-tempb;

	if((tempa-tempb)>0) return tempa-tempb;

	else return tempa-tempb+2*PI;

}

double torad(double deg){return deg/180*PI;}//角度化为弧度

Vector Rotate(Vector a,double rad){//向量逆时针旋转rad弧度

	return Vector(a.x*cos(rad)-a.y*sin(rad),a.x*sin(rad)+a.y*cos(rad));

}

Vector Normal(Vector a){//计算单位法线

	double L=Length(a);

	return Vector(-a.y/L,a.x/L);

}

Point GetLineProjection(Point p,Point a,Point b){//点在直线上的投影

	Vector v=b-a;

	return a+v*(Dot(v,p-a)/Dot(v,v));

}

Point GetLineIntersection(Point p,Vector v,Point q,Vector w){//求直线交点 有唯一交点时可用

	Vector u=p-q;

	double t=Cross(w,u)/Cross(v,w);

	return p+v*t;

}

int ConvexHull(Point* p,int n,Point* sol){//计算凸包

	sort(p,p+n);

	int m=0;

	for(int i=0;i<n;i++){

		while(m>1&&Cross(sol[m-1]-sol[m-2],p[i]-sol[m-2])<=0) m--;

		sol[m++]=p[i];

	}

	int k=m;

	for(int i=n-2;i>=0;i--){

		while(m>k&&Cross(sol[m-1]-sol[m-2],p[i]-sol[m-2])<=0) m--;

		sol[m++]=p[i];

	}

	if(n>0) m--;

	return m;

}

double Heron(double a,double b,double c){//海伦公式

	double p=(a+b+c)/2;

	return sqrt(p*(p-a)*(p-b)*(p-c));

}

bool SegmentProperIntersection(Point a1,Point a2,Point b1,Point b2){//线段规范相交判定

	double c1=Cross(a2-a1,b1-a1),c2=Cross(a2-a1,b2-a1);

	double c3=Cross(b2-b1,a1-b1),c4=Cross(b2-b1,a2-b1);

	return dcmp(c1)*dcmp(c2)<0&&dcmp(c3)*dcmp(c4)<0;

}

double CutConvex(const int n,Point* poly, const Point a,const Point b, vector<Point> result[3]){//有向直线a b 切割凸多边形

	vector<Point> points;

	Point p;

	Point p1=a,p2=b;

	int cur,pre;

	result[0].clear();

	result[1].clear();

	result[2].clear();

	if(n==0) return 0;

	double tempcross;

	tempcross=Cross(p2-p1,poly[0]-p1);

	if(dcmp(tempcross)==0) pre=cur=2;

	else if(tempcross>0) pre=cur=0;

	else pre=cur=1;

	for(int i=0;i<n;i++){

		tempcross=Cross(p2-p1,poly[(i+1)%n]-p1);

		if(dcmp(tempcross)==0) cur=2;

		else if(tempcross>0) cur=0;

		else cur=1;

		if(cur==pre){

			result[cur].push_back(poly[(i+1)%n]);

		}

		else{

			p1=poly[i];

			p2=poly[(i+1)%n];

			p=GetLineIntersection(p1,p2-p1,a,b-a);

			points.push_back(p);

			result[pre].push_back(p);

			result[cur].push_back(p);

			result[cur].push_back(poly[(i+1)%n]);

			pre=cur;

		}

	}

	sort(points.begin(),points.end());

	if(points.size()<2){

		return 0;

	}

	else{

		return Length(points.front()-points.back());

	}

}

double DistanceToSegment(Point p,Segment s){//点到线段的距离

	if(s.a==s.b) return Length(p-s.a);

	Vector v1=s.b-s.a,v2=p-s.a,v3=p-s.b;

	if(dcmp(Dot(v1,v2))<0) return Length(v2);

	else if(dcmp(Dot(v1,v3))>0) return Length(v3);

	else return fabs(Cross(v1,v2))/Length(v1);

}

bool isPointOnSegment(Point p,Segment s){

	return dcmp(Cross(s.a-p,s.b-p))==0&&dcmp(Dot(s.a-p,s.b-p))<0;

}

int isPointInPolygon(Point p, Point* poly,int n){//点与多边形的位置关系

	int wn=0;

	for(int i=0;i<n;i++){

		Point& p2=poly[(i+1)%n];

		if(isPointOnSegment(p,Segment(poly[i],p2))) return -1;//点在边界上

		int k=dcmp(Cross(p2-poly[i],p-poly[i]));

		int d1=dcmp(poly[i].y-p.y);

		int d2=dcmp(p2.y-p.y);

		if(k>0&&d1<=0&&d2>0)wn++;

		if(k<0&&d2<=0&&d1>0)wn--;

	}

	if(wn) return 1;//点在内部

	else return 0;//点在外部

}

double PolygonArea(Point* p,int n){//多边形有向面积

	double area=0;

	for(int i=1;i<n-1;i++)

		area+=Cross(p[i]-p[0],p[i+1]-p[0]);

	return area/2;

}

int GetLineCircleIntersection(Line L,Circle C,Point& p1,Point& p2){//圆与直线交点 返回交点个数

	double a = L.v.x, b = L.p.x - C.c.x, c = L.v.y, d = L.p.y-C.c.y;

	double e = a*a + c*c, f = 2*(a*b+c*d), g = b*b + d*d -C.r*C.r;

	double delta = f*f - 4*e*g;

	if(dcmp(delta) < 0)  return 0;//相离

	if(dcmp(delta) == 0) {//相切

		p1=p1=C.point(-f/(2*e));

		return 1;

	}//相交

	p1=(L.p+L.v*(-f-sqrt(delta))/(2*e));

	p2=(L.p+L.v*(-f+sqrt(delta))/(2*e));

	return 2;

}

double rotating_calipers(Point *ch,int n)//旋转卡壳

{

	int q=1;

	double ans=0;

	ch[n]=ch[0];

	for(int p=0;p<n;p++)

	{

		while(Cross(ch[q+1]-ch[p+1],ch[p]-ch[p+1])>Cross(ch[q]-ch[p+1],ch[p]-ch[p+1]))

			q=(q+1)%n;

		ans=max(ans,max(Length(ch[p]-ch[q]),Length(ch[p+1]-ch[q+1])));

	}

	return ans;

}

Polygon CutPolygon(Polygon poly,Point a,Point b){//用a->b切割多边形 返回左侧

	Polygon newpoly;

	int n=poly.size();

	for(int i=0;i<n;i++){

		Point c=poly[i];

		Point d=poly[(i+1)%n];

		if(dcmp(Cross(b-a,c-a))>=0) newpoly.push_back(c);

		if(dcmp(Cross(b-a,c-d))!=0){

			Point ip=GetLineIntersection(a,b-a,c,d-c);

			if(isPointOnSegment(ip,Segment(c,d))) newpoly.push_back(ip);

		}

	}

	return newpoly;

}

int GetCircleCircleIntersection(Circle c1,Circle c2,Point& p1,Point& p2){//求两圆相交

	double d=Length(c1.c-c2.c);

	if(dcmp(d)==0){

		if(dcmp(c1.r-c2.r)==0) return -1;//两圆重合

		return 0;

	}

	if(dcmp(c1.r+c2.r-d)<0) return 0;

	if(dcmp(fabs(c1.r-c2.r)-d)>0) return 0;

	double a=Angle(c2.c-c1.c,Vector(1,0));

	double da=acos((c1.r*c1.r+d*d-c2.r*c2.r)/(2*c1.r*d));

	p1=c1.point(a-da);p2=c1.point(a+da);

	if(p1==p2) return 1;

	return 2;

}

bool isPointOnleft(Point p,Line L){return dcmp(Cross(L.v,p-L.p))>0;}

int HalfplaneIntersection(Line *L,int n,Point* poly){//半平面交

	sort(L,L+n);

	int first,last;

	Point* p=new Point[n];

	Line* q=new Line[n];

	q[first=last=0]=L[0];

	for(int i=1;i<n;i++){

		while(first<last&&!isPointOnleft(p[last-1],L[i])) last--;

		while(first<last&&!isPointOnleft(p[first],L[i])) first++;

		q[++last]=L[i];

		if(dcmp(Cross(q[last].v,q[last-1].v))==0){

			last--;

			if(isPointOnleft(L[i].p,q[last])) q[last]=L[i];

		}

		if(first<last) p[last-1]=GetLineIntersection(q[last-1].p,q[last-1].v,q[last].p,q[last].v);

	}

	while(first<last&&!isPointOnleft(p[last-1],q[first])) last--;

	if(last-first<=1) return 0;

	p[last]=GetLineIntersection(q[last].p,q[last].v,q[first].p,q[first].v);

	int m=0;

	for(int i=first;i<=last;i++) poly[m++]=p[i];

	return m;

}

//两点式化为一般式A = b.y-a.y, B = a.x-b.x, C = -a.y*(B)-a.x*(A);

//--------------------------------------

//--------------------------------------

//--------------------------------------

//--------------------------------------

//--------------------------------------

Point p[200],poly[200];

Line L[200];

Vector v[200],v2[200];

int main(){

	int n;

	while(scanf("%d",&n)&&n){

		int m,x,y;

		for(int i=0;i<n;i++)

			scanf("%lf%lf",&p[i].x,&p[i].y);

		for(int i=0;i<n;i++){

			v[i]=p[(i+1)%n]-p[i];

			v2[i]=Normal(v[i]);

		}

		double left=0,right=20000;

		while(dcmp(right-left)>0){

			double mid=left+(right-left)/2;

			for(int i=0;i<n;i++) L[i]=Line(p[i]+v2[i]*mid,v[i]);

			m=HalfplaneIntersection(L,n,poly);

			if(!m) right=mid;

			else left=mid;

		}

		printf("%.6lf\n",left);

	}

	return 0;

}
巴特西

1396 - Most Distant Point from the Sea

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