计算几何笔记 （模板）

正弦定理:

(d_{circumcircle}= frac{a}{sin A}=frac{b}{sin B}=frac{c}{sin C})

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余弦定理:

(a^2=b^2+c^2-2bc imes cos A)

(b^2=a^2+c^2-2ac imes cos B)

(c^2=a^2+b^2-2ab imes cos C)

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向量旋转:

(egin{bmatrix}cos heta &-sin heta\sin heta&cos hetaend{bmatrix} imes egin{bmatrix}x\yend{bmatrix}=egin{bmatrix}x^prime\y^primeend{bmatrix})

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向量基本操作:

#include<iostream>
#include<cstdio>
#include<cstring>
#include<vector>
#include<cmath>
#include<algorithm>
#define N 100010
using namespace std;
const double eps=1e-10;
const double PI=acos(-1.0);
//Vector可以用Point来表示 
struct Point{
	double x,y;
	Point(){} Point(double _x,double _y){x=_x,y=_y;}
	//相等
	inline bool operator ==(Point b){
		return fabs(x-b.x)<=eps&&fabs(y-b.y)<=eps;
	} 
	//不相等 
	inline bool operator !=(Point b){
		return !((*this)==b);
	} 
	//相加 
	inline Point operator +(Point b){
		return Point(x+b.x,y+b.y);
	}
	//相减 
	inline Point operator -(Point b){
		return Point(x-b.x,y-b.y);
	}
	//点积 
	inline double operator *(Point b){
		return x*b.x+y*b.y;
	}
	//叉积 
	inline double operator %(Point b){
		return x*b.y-y*b.x;
	}
	//长度 
	inline double len(){
		return sqrt(x*x+y*y);
	}
	//单位向量
	inline Point unit(){
		double tmp=len();
		return Point(x/tmp,y/tmp);
	} 
	//求单位法向量 
	inline Point normal(){
		double tmp=len();
		return Point(-y/tmp,x/tmp);
	}
	//点从下往上排序
	inline bool operator <(Point b){
		return fabs(y-b.y)<eps?x<b.x:y<b.y; 
	} 
	//平面直角坐标系上半部分优先
	inline bool Quad(){
		return y>eps||(fabs(y)<=eps&&x>=-eps);
	} 
}LTL;//Lower Then Left
inline Point operator *(double k,Point a){
	return Point(a.x*k,a.y*k);
}

//两个向量的夹角 
inline double Angle(Point a,Point b){
	//return acos(a*b/a.len()/b.len());
	double tmp=atan2(b.y,b.x)-atan2(a.y,a.x); 
	while(tmp>=PI+eps)tmp-=2*PI;
	while(tmp<=-PI-eps)tmp+=2*PI;
	return tmp;
}
//判断两个向量是否平行
inline bool Para(Point a,Point b){
	return fabs(a%b)<=eps;
} 
//判断向量a是否在向量b的左侧
inline bool Left(Point a,Point b){
	return b%a>eps;
} 
//逆时针旋转p后的向量 
inline Point Rotate(Point a,double p){
	return Point(a.x*cos(p)-a.y*sin(p),a.x*sin(p)+a.y*cos(p));
}
//绕LTL逆时针排序 
inline bool cmp(Point a,Point b){
	a=a-LTL,b=b-LTL;
	return Para(a,b)?a.len()<b.len():Left(b,a);
} 

直线:

//直线 
struct Line{
	Point p[2];
	Line(){} Line(Point a,Point b){p[0]=a,p[1]=b;}
	inline Point& operator [] (int i){
		return p[i];
	}
	inline Point v(){
		return p[1]-p[0];
	}
};
//直线交点 
inline Point Cross(Line a,Line b){
	return a[0]+((b[0]-a[0])%(b[1]-a[0]))/((a[1]-a[0])%(b[1]-b[0]))*(a[1]-a[0]);
} 

多边形:

//多边形 
typedef vector<Point> Poly;
//面积 
inline double Area(Poly a){
	double tot=0;
	for(int i=0;i<a.size()-1;i++){
		tot+=a[i]%a[i+1];
	}
	tot+=a[a.size()-1]%a[0];
	return fabs(tot)/2.0;
} 
//周长
inline double C(Poly a){
	double tot=0;
	for(int i=0;i<a.size()-1;i++){
		tot+=(a[i+1]-a[i]).len();
	}
	tot+=(a[a.size()-1]-a[0]).len();
	return tot;
} 
//重心 
inline Point Cent(Poly a){
	double x=0,y=0,m=0;
	for(int i=1;i<a.size()-1;++i){
		double tmp=(a[i]-a[0])%(a[i+1]-a[0]);
		x+=tmp*(a[0].x+a[i].x+a[i+1].x)/3.0;
		y+=tmp*(a[0].y+a[i].y+a[i+1].y)/3.0;
		m+=tmp;
	}
	return Point(x/m,y/m);
} 

凸包:

//求凸包 (的周长) 
Point st[123456];
int top;
inline double Convex(Poly a){
	top=0;
	LTL=(*min_element(a.begin(),a.end()));
	sort(a.begin(),a.end(),cmp);
	for(int i=0;i<a.size();++i){
		while(top>1&&!Left(a[i]-st[top-1],st[top]-st[top-1]))--top;
		st[++top]=a[i];
	}
	Poly tmp;
	for(int i=1;i<=top;i++)tmp.push_back(st[i]);
	return C(tmp); 
}

点与多边形关系:

// 求点是否在多边形内(单次询问,可以是凹多边形)
inline bool Inside_1(Poly a,Point p){
	double tot=0;
	for(int i=0;i<a.size();++i)tot+=Angle(a[i]-p,a[(i+1)%n]-p);
	return fabs(tot)>=eps;
} 

//多组询问
inline bool Inside_2(Poly &a,Point p){
	if(p==LTL)return true;
	if(Left(p-LTL,a[a.size()-1]-LTL)||Left(a[1]-LTL,p-LTL))return false; 
	int pos=lower_bound(a.begin(),a.end()-1,p,cmp)-a.begin();
	return !Left(a[pos]-a[pos-1],p-a[pos-1])&&LTL<p;
}

半平面交:

//半平面交(HalfPlaneIntersection)
inline bool cmpang(Point a,Point b){
	return a.Quad()^b.Quad()?a.Quad()<b.Quad():Left(b,a);
}
inline bool operator <(Line a,Line b){
	return Para(a.v(),b.v())&&(a.v()*b.v()>eps)?Left(a[0]-b[0],b.v()):cmpang(a.v(),b.v());
}
Line q[N];
int l,r;
Poly HPI(Line L[],int m){
	sort(L+1,L+m+1);
	Poly R;
	r=1,q[l=1]=L[1];
	for(int i=2;i<=m;i++){
		if(!cmpang(q[r].v(),L[i].v()))continue;
		while(l<r&&Left(L[i].v(),Cross(q[r],q[r-1])-L[i][0]))--r;
		while(l<r&&Left(L[i].v(),Cross(q[l],q[l+1])-L[i][0]))++l;
		q[++r]=L[i];
	}
	while(l<r&&Left(q[l].v(),Cross(q[r],q[r-1])-q[l][0]))--r;
	while(l<r&&Left(q[r].v(),Cross(q[l],q[l+1])-q[r][0]))++l;
	q[r+1]=q[l];
	for(int i=l;i<=r;++i){
		R.push_back(Cross(q[i],q[i+1]));
	}
	return R;
}

Minkowski和:

//Minkowski
Poly Minkowski(Poly a,Poly b){
	Poly R;
	int i=0,j=0;
	R.push_back(a[0]+b[0]);
	while(1){
		if(Left(a[(i+1)%a.size()]-a[i],b[(j+1)%b.size()]-b[j])){
			j=(j+1)%b.size();
		}
		else{
			i=(i+1)%a.size();
		}
		Point tmp=a[i]+b[j];
		if(R.size()>1&&Para(tmp-R[R.size()-1],R[R.size()-1]-R[R.size()-2])){
			R.pop_back();
		}
		R.push_back(tmp);
		if(!i&&!j)break;
	}
	R.pop_back();
	return R;
}

旋转卡壳:

int RC(Poly a){
	int ans=0,j=1;
	a.push_back(a[0]);
	for(int i=0;i<a.size()-1;++i){
		while((a[i]-a[j])%(a[i+1]-a[j])<(a[i]-a[j+1])%(a[i+1]-a[j+1]))j=(j+1)%a.size();
		ans=max(ans,max((a[j]-a[i]).len(),(a[j]-a[i+1]).len()));
	}
	return ans;
}