#include <iostream>
#include <math.h>
#define eps 1e-8
#define zero(x) (((x)>0?(x):-(x))<eps)
#define pi acos(-1.0)
struct point
{
double x,y;
};
struct line
{
point a,b;
};
struct point3
{
double x,y,z;
};
struct line3
{
point3 a,b;
};
struct plane3
{
point3 a,b,c;
};
//計算cross product (P1-P0)x(P2-P0)
double xmult(point p1,point p2,point p0)
{
return (p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
//計算dot product (P1-P0).(P2-P0)
double dmult(point p1,point p2,point p0)
{
return (p1.x-p0.x)*(p2.x-p0.x)+(p1.y-p0.y)*(p2.y-p0.y);
}
//計算cross product U . V
point3 xmult(point3 u,point3 v)
{
point3 ret;
ret.x=u.y*v.z-v.y*u.z;
ret.y=u.z*v.x-u.x*v.z;
ret.z=u.x*v.y-u.y*v.x;
return ret;
}
//計算dot product U . V
double dmult(point3 u,point3 v)
{
return u.x*v.x+u.y*v.y+u.z*v.z;
}
//兩點距離
double distance(point p1,point p2)
{
return sqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));
}
//判三點共線
bool dots_inline(point p1,point p2,point p3)
{
return zero(xmult(p1,p2,p3));
}
//判點是否在線段上,包括端點
bool dot_online_in(point p,line l)
{
return zero(xmult(p,l.a,l.b))&&(l.a.x-p.x)*(l.b.x-p.x)<eps&&(l.a.y-p.y)*(l.b.y-p.y)<eps;
}
//判點是否在線段上,不包括端點
bool dot_online_ex(point p,line l)
{
return dot_online_in(p,l)&&(!zero(p.x-l.a.x)||!zero(p.y-l.a.y))&&(!zero(p.x-l.b.x)||!zero(p.y-l.b.y));
}
//判兩點在線段同側,點在線段上返回0
bool same_side(point p1,point p2,line l)
{
return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)>eps;
}
//判兩點在線段異側,點在線段上返回0
bool opposite_side(point p1,point p2,line l)
{
return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)<-eps;
}
//判兩直線平行
bool parallel(line u,line v)
{
return zero((u.a.x-u.b.x)*(v.a.y-v.b.y)-(v.a.x-v.b.x)*(u.a.y-u.b.y));
}
//判兩直線垂直
bool perpendicular(line u,line v)
{
return zero((u.a.x-u.b.x)*(v.a.x-v.b.x)+(u.a.y-u.b.y)*(v.a.y-v.b.y));
}
//判兩線段相交,包括端點和部分重合
bool intersect_in(line u,line v)
{
if (!dots_inline(u.a,u.b,v.a)||!dots_inline(u.a,u.b,v.b))
return !same_side(u.a,u.b,v)&&!same_side(v.a,v.b,u);
return dot_online_in(u.a,v)||dot_online_in(u.b,v)||dot_online_in(v.a,u)||dot_online_in(v.b,u);
}
//判兩線段相交,不包括端點和部分重合
bool intersect_ex(line u,line v)
{
return opposite_side(u.a,u.b,v)&&opposite_side(v.a,v.b,u);
}
//計算兩直線交點,注意事先判斷直線是否平行!
//線段交點請另外判線段相交(同時還是要判斷是否平行!)
point intersection(line u,line v)
{
point ret=u.a;
double t=((u.a.x-v.a.x)*(v.a.y-v.b.y)-(u.a.y-v.a.y)*(v.a.x-v.b.x))
/((u.a.x-u.b.x)*(v.a.y-v.b.y)-(u.a.y-u.b.y)*(v.a.x-v.b.x));
ret.x+=(u.b.x-u.a.x)*t;
ret.y+=(u.b.y-u.a.y)*t;
return ret;
}
point intersection(point u1,point u2,point v1,point v2)
{
point ret=u1;
double t=((u1.x-v1.x)*(v1.y-v2.y)-(u1.y-v1.y)*(v1.x-v2.x))
/((u1.x-u2.x)*(v1.y-v2.y)-(u1.y-u2.y)*(v1.x-v2.x));
ret.x+=(u2.x-u1.x)*t;
ret.y+=(u2.y-u1.y)*t;
return ret;
}
//點到直線上的最近點
point ptoline(point p,line l)
{
point t=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
return intersection(p,t,l.a,l.b);
}
//點到直線距離
double disptoline(point p,line l)
{
return fabs(xmult(p,l.a,l.b))/distance(l.a,l.b);
}
//點到線段上的最近點
point ptoseg(point p,line l)
{
point t=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
if (xmult(l.a,t,p)*xmult(l.b,t,p)>eps)
return distance(p,l.a)<distance(p,l.b)?l.a:l.b;
return intersection(p,t,l.a,l.b);
}
//點到線段距離
double disptoseg(point p,line l)
{
point t=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
if (xmult(l.a,t,p)*xmult(l.b,t,p)>eps)
return distance(p,l.a)<distance(p,l.b)?distance(p,l.a):distance(p,l.b);
return fabs(xmult(p,l.a,l.b))/distance(l.a,l.b);
}
//矢量V 以P 為頂點逆時針旋轉angle 並放大scale 倍
point rotate(point v,point p,double angle,double scale)
{
point ret=p;
v.x-=p.x,v.y-=p.y;
p.x=scale*cos(angle);
p.y=scale*sin(angle);
ret.x+=v.x*p.x-v.y*p.y;
ret.y+=v.x*p.y+v.y*p.x;
return ret;
}
//計算三角形面積,輸入三頂點
double area_triangle(point p1,point p2,point p3)
{
return fabs(xmult(p1,p2,p3))/2;
}
//計算三角形面積,輸入三邊長
double area_triangle(double a,double b,double c)
{
double s=(a+b+c)/2;
return sqrt(s*(s-a)*(s-b)*(s-c));
}
//計算多邊形面積,頂點按順時針或逆時針給出
double area_polygon(int n,point* p)
{
double s1=0,s2=0;
int i;
for (i=0; i<n; i++)
s1+=p[(i+1)%n].y*p[i].x,s2+=p[(i+1)%n].y*p[(i+2)%n].x;
return fabs(s1-s2)/2;
}
//計算圓心角lat 表示緯度,-90<=w<=90,lng 表示經度
//返回兩點所在大圓劣弧對應圓心角,0<=angle<=pi
double angle(double lng1,double lat1,double lng2,double lat2)
{
double dlng=fabs(lng1-lng2)*pi/180;
while (dlng>=pi+pi)
dlng-=pi+pi;
if (dlng>pi)
dlng=pi+pi-dlng;
lat1*=pi/180,lat2*=pi/180;
return acos(cos(lat1)*cos(lat2)*cos(dlng)+sin(lat1)*sin(lat2));
}
//計算距離,r 為球半徑
double line_dist(double r,double lng1,double lat1,double lng2,double lat2)
{
double dlng=fabs(lng1-lng2)*pi/180;
while (dlng>=pi+pi)
dlng-=pi+pi;
if (dlng>pi)
dlng=pi+pi-dlng;
lat1*=pi/180,lat2*=pi/180;
return r*sqrt(2-2*(cos(lat1)*cos(lat2)*cos(dlng)+sin(lat1)*sin(lat2)));
}
//計算球面距離,r 為球半徑
inline double sphere_dist(double r,double lng1,double lat1,double lng2,double lat2)
{
return r*angle(lng1,lat1,lng2,lat2);
}
//外心
point circumcenter(point a,point b,point c)
{
line u,v;
u.a.x=(a.x+b.x)/2;
u.a.y=(a.y+b.y)/2;
u.b.x=u.a.x-a.y+b.y;
u.b.y=u.a.y+a.x-b.x;
v.a.x=(a.x+c.x)/2;
v.a.y=(a.y+c.y)/2;
v.b.x=v.a.x-a.y+c.y;
v.b.y=v.a.y+a.x-c.x;
return intersection(u,v);
}
//內心
point incenter(point a,point b,point c)
{
line u,v;
double m,n;
u.a=a;
m=atan2(b.y-a.y,b.x-a.x);
n=atan2(c.y-a.y,c.x-a.x);
u.b.x=u.a.x+cos((m+n)/2);
u.b.y=u.a.y+sin((m+n)/2);
v.a=b;
m=atan2(a.y-b.y,a.x-b.x);
n=atan2(c.y-b.y,c.x-b.x);
v.b.x=v.a.x+cos((m+n)/2);
v.b.y=v.a.y+sin((m+n)/2);
return intersection(u,v);
}
//垂心
point perpencenter(point a,point b,point c)
{
line u,v;
u.a=c;
u.b.x=u.a.x-a.y+b.y;
u.b.y=u.a.y+a.x-b.x;
v.a=b;
v.b.x=v.a.x-a.y+c.y;
v.b.y=v.a.y+a.x-c.x;
return intersection(u,v);
}
//重心
//到三角形三頂點距離的平方和最小的點
//三角形內到三邊距離之積最大的點
point barycenter(point a,point b,point c)
{
line u,v;
u.a.x=(a.x+b.x)/2;
u.a.y=(a.y+b.y)/2;
u.b=c;
v.a.x=(a.x+c.x)/2;
v.a.y=(a.y+c.y)/2;
v.b=b;
return intersection(u,v);
}
//費馬點
//到三角形三頂點距離之和最小的點
point fermentpoint(point a,point b,point c)
{
point u,v;
double step=fabs(a.x)+fabs(a.y)+fabs(b.x)+fabs(b.y)+fabs(c.x)+fabs(c.y);
int i,j,k;
u.x=(a.x+b.x+c.x)/3;
u.y=(a.y+b.y+c.y)/3;
while (step>1e-10)
{
for (k=0; k<10; step/=2,k++)
{
for (i=-1; i<=1; i++)
{
for (j=-1; j<=1; j++)
{
v.x=u.x+step*i;
v.y=u.y+step*j;
if(distance(u,a)+distance(u,b)+distance(u,c)>distance(v,a)+distance(v,b)+distance(v,c))
{
u=v;
}
}
}
}
}
return u;
}
//矢量差 U - V
point3 subt(point3 u,point3 v)
{
point3 ret;
ret.x=u.x-v.x;
ret.y=u.y-v.y;
ret.z=u.z-v.z;
return ret;
}
//取平面法向量
point3 pvec(plane3 s)
{
return xmult(subt(s.a,s.b),subt(s.b,s.c));
}
point3 pvec(point3 s1,point3 s2,point3 s3)
{
return xmult(subt(s1,s2),subt(s2,s3));
}
//兩點距離,單參數取向量大小
double distance(point3 p1,point3 p2)
{
return sqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y)+(p1.z-p2.z)*(p1.z-p2.z));
}
///三維///
//向量大小
double vlen(point3 p)
{
return sqrt(p.x*p.x+p.y*p.y+p.z*p.z);
}
//判三點共線
bool dots_inline(point3 p1,point3 p2,point3 p3)
{
return vlen(xmult(subt(p1,p2),subt(p2,p3)))<eps;
}
//判四點共面
bool dots_onplane(point3 a,point3 b,point3 c,point3 d)
{
return zero(dmult(pvec(a,b,c),subt(d,a)));
}
//判點是否在線段上,包括端點和共線
bool dot_online_in(point3 p,line3 l)
{
return zero(vlen(xmult(subt(p,l.a),subt(p,l.b))))&&(l.a.x-p.x)*(l.b.x-p.x)<eps&&(l.a.y-p.y)*(l.b.y-p.y)<eps&&(l.a.z-p.z)*(l.b.z-p.z)<eps;
}
//判點是否在線段上,不包括端點
bool dot_online_ex(point3 p,line3 l)
{
return dot_online_in(p,l)&&(!zero(p.x-l.a.x)||!zero(p.y-l.a.y)||!zero(p.z-l.a.z))&&(!zero(p.x-l.b.x)||!zero(p.y-l.b.y)||!zero(p.z-l.b.z));
}
//判點是否在空間三角形上,包括邊界,三點共線無意義
bool dot_inplane_in(point3 p,plane3 s)
{
return zero(vlen(xmult(subt(s.a,s.b),subt(s.a,s.c)))-vlen(xmult(subt(p,s.a),subt(p,s.b)))-vlen(xmult(subt(p,s.b),subt(p,s.c)))-vlen(xmult(subt(p,s.c),subt(p,s.a))));
}
//判點是否在空間三角形上,不包括邊界,三點共線無意義
bool dot_inplane_ex(point3 p,plane3 s)
{
return dot_inplane_in(p,s)&&vlen(xmult(subt(p,s.a),subt(p,s.b)))>eps&&vlen(xmult(subt(p,s.b),subt(p,s.c)))>eps&&vlen(xmult(subt(p,s.c),subt(p,s.a)))>eps;
}
//判兩點在線段同側,點在線段上返回0,不共面無意義
bool same_side(point3 p1,point3 p2,line3 l)
{
return dmult(xmult(subt(l.a,l.b),subt(p1,l.b)),xmult(subt(l.a,l.b),subt(p2,l.b)))>eps;
}
//判兩點在線段異側,點在線段上返回0,不共面無意義
bool opposite_side(point3 p1,point3 p2,line3 l)
{
return dmult(xmult(subt(l.a,l.b),subt(p1,l.b)),xmult(subt(l.a,l.b),subt(p2,l.b)))<-eps;
}
//判兩點在平面同側,點在平面上返回0
bool same_side(point3 p1,point3 p2,plane3 s)
{
return dmult(pvec(s),subt(p1,s.a))*dmult(pvec(s),subt(p2,s.a))>eps;
}
bool same_side(point3 p1,point3 p2,point3 s1,point3 s2,point3 s3)
{
return dmult(pvec(s1,s2,s3),subt(p1,s1))*dmult(pvec(s1,s2,s3),subt(p2,s1))>eps;
}
//判兩點在平面異側,點在平面上返回0
bool opposite_side(point3 p1,point3 p2,plane3 s)
{
return dmult(pvec(s),subt(p1,s.a))*dmult(pvec(s),subt(p2,s.a))<-eps;
}
bool opposite_side(point3 p1,point3 p2,point3 s1,point3 s2,point3 s3)
{
return dmult(pvec(s1,s2,s3),subt(p1,s1))*dmult(pvec(s1,s2,s3),subt(p2,s1))<-eps;
}
//判兩直線平行
bool parallel(line3 u,line3 v)
{
return vlen(xmult(subt(u.a,u.b),subt(v.a,v.b)))<eps;
}
//判兩平面平行
bool parallel(plane3 u,plane3 v)
{
return vlen(xmult(pvec(u),pvec(v)))<eps;
}
//判直線與平面平行
bool parallel(line3 l,plane3 s)
{
return zero(dmult(subt(l.a,l.b),pvec(s)));
}
bool parallel(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3)
{
return zero(dmult(subt(l1,l2),pvec(s1,s2,s3)));
}
//判兩直線垂直
bool perpendicular(line3 u,line3 v)
{
return zero(dmult(subt(u.a,u.b),subt(v.a,v.b)));
}
//判兩平面垂直
bool perpendicular(plane3 u,plane3 v)
{
return zero(dmult(pvec(u),pvec(v)));
}
//判直線與平面平行
bool perpendicular(line3 l,plane3 s)
{
return vlen(xmult(subt(l.a,l.b),pvec(s)))<eps;
}
//判兩線段相交,包括端點和部分重合
bool intersect_in(line3 u,line3 v)
{
if (!dots_onplane(u.a,u.b,v.a,v.b))
return 0;
if (!dots_inline(u.a,u.b,v.a)||!dots_inline(u.a,u.b,v.b))
return !same_side(u.a,u.b,v)&&!same_side(v.a,v.b,u);
return dot_online_in(u.a,v)||dot_online_in(u.b,v)||dot_online_in(v.a,u)||dot_online_in(v.b,u);
}
//判兩線段相交,不包括端點和部分重合
bool intersect_ex(line3 u,line3 v)
{
return dots_onplane(u.a,u.b,v.a,v.b)&&opposite_side(u.a,u.b,v)&&opposite_side(v.a,v.b,u);
}
//判線段與空間三角形相交,包括交於邊界和(部分)包含
bool intersect_in(line3 l,plane3 s)
{
return !same_side(l.a,l.b,s)&&!same_side(s.a,s.b,l.a,l.b,s.c)&&!same_side(s.b,s.c,l.a,l.b,s.a)&&!same_side(s.c,s.a,l.a,l.b,s.b);
}
//判線段與空間三角形相交,不包括交於邊界和(部分)包含
bool intersect_ex(line3 l,plane3 s)
{
return opposite_side(l.a,l.b,s)&&opposite_side(s.a,s.b,l.a,l.b,s.c)&&opposite_side(s.b,s.c,l.a,l.b,s.a)&&opposite_side(s.c,s.a,l.a,l.b,s.b);
}
//計算兩直線交點,注意事先判斷直線是否共面和平行!
//線段交點請另外判線段相交(同時還是要判斷是否平行!)
point3 intersection(line3 u,line3 v)
{
point3 ret=u.a;
double t=((u.a.x-v.a.x)*(v.a.y-v.b.y)-(u.a.y-v.a.y)*(v.a.x-v.b.x))
/((u.a.x-u.b.x)*(v.a.y-v.b.y)-(u.a.y-u.b.y)*(v.a.x-v.b.x));
ret.x+=(u.b.x-u.a.x)*t;
ret.y+=(u.b.y-u.a.y)*t;
ret.z+=(u.b.z-u.a.z)*t;
return ret;
}
//計算直線與平面交點,注意事先判斷是否平行,並保證三點不共線!
//線段和空間三角形交點請另外判斷
point3 intersection(line3 l,plane3 s)
{
point3 ret=pvec(s);
double t=(ret.x*(s.a.x-l.a.x)+ret.y*(s.a.y-l.a.y)+ret.z*(s.a.z-l.a.z))/(ret.x*(l.b.x-l.a.x)+ret.y*(l.b.y-l.a.y)+ret.z*(l.b.z-l.a.z));
ret.x=l.a.x+(l.b.x-l.a.x)*t;
ret.y=l.a.y+(l.b.y-l.a.y)*t;
ret.z=l.a.z+(l.b.z-l.a.z)*t;
return ret;
}
point3 intersection(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3)
{
point3 ret=pvec(s1,s2,s3);
double t=(ret.x*(s1.x-l1.x)+ret.y*(s1.y-l1.y)+ret.z*(s1.z-l1.z))/
(ret.x*(l2.x-l1.x)+ret.y*(l2.y-l1.y)+ret.z*(l2.z-l1.z));
ret.x=l1.x+(l2.x-l1.x)*t;
ret.y=l1.y+(l2.y-l1.y)*t;
ret.z=l1.z+(l2.z-l1.z)*t;
return ret;
}
//計算兩平面交線,注意事先判斷是否平行,並保證三點不共線!
line3 intersection(plane3 u,plane3 v)
{
line3 ret;
ret.a=parallel(v.a,v.b,u.a,u.b,u.c)?intersection(v.b,v.c,u.a,u.b,u.c):intersection(v.a,v.b,u.a,u.b,u.c);
ret.b=parallel(v.c,v.a,u.a,u.b,u.c)?intersection(v.b,v.c,u.a,u.b,u.c):intersection(v.c,v.a,u.a,u.b,u.c);
return ret;
}
line3 intersection(point3 u1,point3 u2,point3 u3,point3 v1,point3 v2,point3 v3)
{
line3 ret;
ret.a=parallel(v1,v2,u1,u2,u3)?intersection(v2,v3,u1,u2,u3):intersection(v1,v2,u1,u2,u3);
ret.b=parallel(v3,v1,u1,u2,u3)?intersection(v2,v3,u1,u2,u3):intersection(v3,v1,u1,u2,u3);
return ret;
}
//點到直線距離
double ptoline(point3 p,line3 l)
{
return vlen(xmult(subt(p,l.a),subt(l.b,l.a)))/distance(l.a,l.b);
}
//點到平面距離
double ptoplane(point3 p,plane3 s)
{
return fabs(dmult(pvec(s),subt(p,s.a)))/vlen(pvec(s));
}
//直線到直線距離
double linetoline(line3 u,line3 v)
{
point3 n=xmult(subt(u.a,u.b),subt(v.a,v.b));
return fabs(dmult(subt(u.a,v.a),n))/vlen(n);
}
//兩直線夾角cos 值
double angle_cos(line3 u,line3 v)
{
return dmult(subt(u.a,u.b),subt(v.a,v.b))/vlen(subt(u.a,u.b))/vlen(subt(v.a,v.b));
}
//兩平面夾角cos 值
double angle_cos(plane3 u,plane3 v)
{
return dmult(pvec(u),pvec(v))/vlen(pvec(u))/vlen(pvec(v));
}
//直線平面夾角sin 值
double angle_sin(line3 l,plane3 s)
{
return dmult(subt(l.a,l.b),pvec(s))/vlen(subt(l.a,l.b))/vlen(pvec(s));
}
int main()
{
return 0;
}
#include <iostream>
#include <math.h>
#define eps 1e-8
#define zero(x) (((x)>0?(x):-(x))<eps)
#define pi acos(-1.0)
struct point
{
double x,y;
};
struct line
{
point a,b;
};
struct point3
{
double x,y,z;
};
struct line3
{
point3 a,b;
};
struct plane3
{
point3 a,b,c;
};
//計算cross product (P1-P0)x(P2-P0)
double xmult(point p1,point p2,point p0)
{
return (p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
//計算dot product (P1-P0).(P2-P0)
double dmult(point p1,point p2,point p0)
{
return (p1.x-p0.x)*(p2.x-p0.x)+(p1.y-p0.y)*(p2.y-p0.y);
}
//計算cross product U . V
point3 xmult(point3 u,point3 v)
{
point3 ret;
ret.x=u.y*v.z-v.y*u.z;
ret.y=u.z*v.x-u.x*v.z;
ret.z=u.x*v.y-u.y*v.x;
return ret;
}
//計算dot product U . V
double dmult(point3 u,point3 v)
{
return u.x*v.x+u.y*v.y+u.z*v.z;
}
//兩點距離
double distance(point p1,point p2)
{
return sqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));
}
//判三點共線
bool dots_inline(point p1,point p2,point p3)
{
return zero(xmult(p1,p2,p3));
}
//判點是否在線段上,包括端點
bool dot_online_in(point p,line l)
{
return zero(xmult(p,l.a,l.b))&&(l.a.x-p.x)*(l.b.x-p.x)<eps&&(l.a.y-p.y)*(l.b.y-p.y)<eps;
}
//判點是否在線段上,不包括端點
bool dot_online_ex(point p,line l)
{
return dot_online_in(p,l)&&(!zero(p.x-l.a.x)||!zero(p.y-l.a.y))&&(!zero(p.x-l.b.x)||!zero(p.y-l.b.y));
}
//判兩點在線段同側,點在線段上返回0
bool same_side(point p1,point p2,line l)
{
return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)>eps;
}
//判兩點在線段異側,點在線段上返回0
bool opposite_side(point p1,point p2,line l)
{
return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)<-eps;
}
//判兩直線平行
bool parallel(line u,line v)
{
return zero((u.a.x-u.b.x)*(v.a.y-v.b.y)-(v.a.x-v.b.x)*(u.a.y-u.b.y));
}
//判兩直線垂直
bool perpendicular(line u,line v)
{
return zero((u.a.x-u.b.x)*(v.a.x-v.b.x)+(u.a.y-u.b.y)*(v.a.y-v.b.y));
}
//判兩線段相交,包括端點和部分重合
bool intersect_in(line u,line v)
{
if (!dots_inline(u.a,u.b,v.a)||!dots_inline(u.a,u.b,v.b))
return !same_side(u.a,u.b,v)&&!same_side(v.a,v.b,u);
return dot_online_in(u.a,v)||dot_online_in(u.b,v)||dot_online_in(v.a,u)||dot_online_in(v.b,u);
}
//判兩線段相交,不包括端點和部分重合
bool intersect_ex(line u,line v)
{
return opposite_side(u.a,u.b,v)&&opposite_side(v.a,v.b,u);
}
//計算兩直線交點,注意事先判斷直線是否平行!
//線段交點請另外判線段相交(同時還是要判斷是否平行!)
point intersection(line u,line v)
{
point ret=u.a;
double t=((u.a.x-v.a.x)*(v.a.y-v.b.y)-(u.a.y-v.a.y)*(v.a.x-v.b.x))
/((u.a.x-u.b.x)*(v.a.y-v.b.y)-(u.a.y-u.b.y)*(v.a.x-v.b.x));
ret.x+=(u.b.x-u.a.x)*t;
ret.y+=(u.b.y-u.a.y)*t;
return ret;
}
point intersection(point u1,point u2,point v1,point v2)
{
point ret=u1;
double t=((u1.x-v1.x)*(v1.y-v2.y)-(u1.y-v1.y)*(v1.x-v2.x))
/((u1.x-u2.x)*(v1.y-v2.y)-(u1.y-u2.y)*(v1.x-v2.x));
ret.x+=(u2.x-u1.x)*t;
ret.y+=(u2.y-u1.y)*t;
return ret;
}
//點到直線上的最近點
point ptoline(point p,line l)
{
point t=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
return intersection(p,t,l.a,l.b);
}
//點到直線距離
double disptoline(point p,line l)
{
return fabs(xmult(p,l.a,l.b))/distance(l.a,l.b);
}
//點到線段上的最近點
point ptoseg(point p,line l)
{
point t=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
if (xmult(l.a,t,p)*xmult(l.b,t,p)>eps)
return distance(p,l.a)<distance(p,l.b)?l.a:l.b;
return intersection(p,t,l.a,l.b);
}
//點到線段距離
double disptoseg(point p,line l)
{
point t=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
if (xmult(l.a,t,p)*xmult(l.b,t,p)>eps)
return distance(p,l.a)<distance(p,l.b)?distance(p,l.a):distance(p,l.b);
return fabs(xmult(p,l.a,l.b))/distance(l.a,l.b);
}
//矢量V 以P 為頂點逆時針旋轉angle 並放大scale 倍
point rotate(point v,point p,double angle,double scale)
{
point ret=p;
v.x-=p.x,v.y-=p.y;
p.x=scale*cos(angle);
p.y=scale*sin(angle);
ret.x+=v.x*p.x-v.y*p.y;
ret.y+=v.x*p.y+v.y*p.x;
return ret;
}
//計算三角形面積,輸入三頂點
double area_triangle(point p1,point p2,point p3)
{
return fabs(xmult(p1,p2,p3))/2;
}
//計算三角形面積,輸入三邊長
double area_triangle(double a,double b,double c)
{
double s=(a+b+c)/2;
return sqrt(s*(s-a)*(s-b)*(s-c));
}
//計算多邊形面積,頂點按順時針或逆時針給出
double area_polygon(int n,point* p)
{
double s1=0,s2=0;
int i;
for (i=0; i<n; i++)
s1+=p[(i+1)%n].y*p[i].x,s2+=p[(i+1)%n].y*p[(i+2)%n].x;
return fabs(s1-s2)/2;
}
//計算圓心角lat 表示緯度,-90<=w<=90,lng 表示經度
//返回兩點所在大圓劣弧對應圓心角,0<=angle<=pi
double angle(double lng1,double lat1,double lng2,double lat2)
{
double dlng=fabs(lng1-lng2)*pi/180;
while (dlng>=pi+pi)
dlng-=pi+pi;
if (dlng>pi)
dlng=pi+pi-dlng;
lat1*=pi/180,lat2*=pi/180;
return acos(cos(lat1)*cos(lat2)*cos(dlng)+sin(lat1)*sin(lat2));
}
//計算距離,r 為球半徑
double line_dist(double r,double lng1,double lat1,double lng2,double lat2)
{
double dlng=fabs(lng1-lng2)*pi/180;
while (dlng>=pi+pi)
dlng-=pi+pi;
if (dlng>pi)
dlng=pi+pi-dlng;
lat1*=pi/180,lat2*=pi/180;
return r*sqrt(2-2*(cos(lat1)*cos(lat2)*cos(dlng)+sin(lat1)*sin(lat2)));
}
//計算球面距離,r 為球半徑
inline double sphere_dist(double r,double lng1,double lat1,double lng2,double lat2)
{
return r*angle(lng1,lat1,lng2,lat2);
}
//外心
point circumcenter(point a,point b,point c)
{
line u,v;
u.a.x=(a.x+b.x)/2;
u.a.y=(a.y+b.y)/2;
u.b.x=u.a.x-a.y+b.y;
u.b.y=u.a.y+a.x-b.x;
v.a.x=(a.x+c.x)/2;
v.a.y=(a.y+c.y)/2;
v.b.x=v.a.x-a.y+c.y;
v.b.y=v.a.y+a.x-c.x;
return intersection(u,v);
}
//內心
point incenter(point a,point b,point c)
{
line u,v;
double m,n;
u.a=a;
m=atan2(b.y-a.y,b.x-a.x);
n=atan2(c.y-a.y,c.x-a.x);
u.b.x=u.a.x+cos((m+n)/2);
u.b.y=u.a.y+sin((m+n)/2);
v.a=b;
m=atan2(a.y-b.y,a.x-b.x);
n=atan2(c.y-b.y,c.x-b.x);
v.b.x=v.a.x+cos((m+n)/2);
v.b.y=v.a.y+sin((m+n)/2);
return intersection(u,v);
}
//垂心
point perpencenter(point a,point b,point c)
{
line u,v;
u.a=c;
u.b.x=u.a.x-a.y+b.y;
u.b.y=u.a.y+a.x-b.x;
v.a=b;
v.b.x=v.a.x-a.y+c.y;
v.b.y=v.a.y+a.x-c.x;
return intersection(u,v);
}
//重心
//到三角形三頂點距離的平方和最小的點
//三角形內到三邊距離之積最大的點
point barycenter(point a,point b,point c)
{
line u,v;
u.a.x=(a.x+b.x)/2;
u.a.y=(a.y+b.y)/2;
u.b=c;
v.a.x=(a.x+c.x)/2;
v.a.y=(a.y+c.y)/2;
v.b=b;
return intersection(u,v);
}
//費馬點
//到三角形三頂點距離之和最小的點
point fermentpoint(point a,point b,point c)
{
point u,v;
double step=fabs(a.x)+fabs(a.y)+fabs(b.x)+fabs(b.y)+fabs(c.x)+fabs(c.y);
int i,j,k;
u.x=(a.x+b.x+c.x)/3;
u.y=(a.y+b.y+c.y)/3;
while (step>1e-10)
{
for (k=0; k<10; step/=2,k++)
{
for (i=-1; i<=1; i++)
{
for (j=-1; j<=1; j++)
{
v.x=u.x+step*i;
v.y=u.y+step*j;
if(distance(u,a)+distance(u,b)+distance(u,c)>distance(v,a)+distance(v,b)+distance(v,c))
{
u=v;
}
}
}
}
}
return u;
}
//矢量差 U - V
point3 subt(point3 u,point3 v)
{
point3 ret;
ret.x=u.x-v.x;
ret.y=u.y-v.y;
ret.z=u.z-v.z;
return ret;
}
//取平面法向量
point3 pvec(plane3 s)
{
return xmult(subt(s.a,s.b),subt(s.b,s.c));
}
point3 pvec(point3 s1,point3 s2,point3 s3)
{
return xmult(subt(s1,s2),subt(s2,s3));
}
//兩點距離,單參數取向量大小
double distance(point3 p1,point3 p2)
{
return sqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y)+(p1.z-p2.z)*(p1.z-p2.z));
}
///三維///
//向量大小
double vlen(point3 p)
{
return sqrt(p.x*p.x+p.y*p.y+p.z*p.z);
}
//判三點共線
bool dots_inline(point3 p1,point3 p2,point3 p3)
{
return vlen(xmult(subt(p1,p2),subt(p2,p3)))<eps;
}
//判四點共面
bool dots_onplane(point3 a,point3 b,point3 c,point3 d)
{
return zero(dmult(pvec(a,b,c),subt(d,a)));
}
//判點是否在線段上,包括端點和共線
bool dot_online_in(point3 p,line3 l)
{
return zero(vlen(xmult(subt(p,l.a),subt(p,l.b))))&&(l.a.x-p.x)*(l.b.x-p.x)<eps&&(l.a.y-p.y)*(l.b.y-p.y)<eps&&(l.a.z-p.z)*(l.b.z-p.z)<eps;
}
//判點是否在線段上,不包括端點
bool dot_online_ex(point3 p,line3 l)
{
return dot_online_in(p,l)&&(!zero(p.x-l.a.x)||!zero(p.y-l.a.y)||!zero(p.z-l.a.z))&&(!zero(p.x-l.b.x)||!zero(p.y-l.b.y)||!zero(p.z-l.b.z));
}
//判點是否在空間三角形上,包括邊界,三點共線無意義
bool dot_inplane_in(point3 p,plane3 s)
{
return zero(vlen(xmult(subt(s.a,s.b),subt(s.a,s.c)))-vlen(xmult(subt(p,s.a),subt(p,s.b)))-vlen(xmult(subt(p,s.b),subt(p,s.c)))-vlen(xmult(subt(p,s.c),subt(p,s.a))));
}
//判點是否在空間三角形上,不包括邊界,三點共線無意義
bool dot_inplane_ex(point3 p,plane3 s)
{
return dot_inplane_in(p,s)&&vlen(xmult(subt(p,s.a),subt(p,s.b)))>eps&&vlen(xmult(subt(p,s.b),subt(p,s.c)))>eps&&vlen(xmult(subt(p,s.c),subt(p,s.a)))>eps;
}
//判兩點在線段同側,點在線段上返回0,不共面無意義
bool same_side(point3 p1,point3 p2,line3 l)
{
return dmult(xmult(subt(l.a,l.b),subt(p1,l.b)),xmult(subt(l.a,l.b),subt(p2,l.b)))>eps;
}
//判兩點在線段異側,點在線段上返回0,不共面無意義
bool opposite_side(point3 p1,point3 p2,line3 l)
{
return dmult(xmult(subt(l.a,l.b),subt(p1,l.b)),xmult(subt(l.a,l.b),subt(p2,l.b)))<-eps;
}
//判兩點在平面同側,點在平面上返回0
bool same_side(point3 p1,point3 p2,plane3 s)
{
return dmult(pvec(s),subt(p1,s.a))*dmult(pvec(s),subt(p2,s.a))>eps;
}
bool same_side(point3 p1,point3 p2,point3 s1,point3 s2,point3 s3)
{
return dmult(pvec(s1,s2,s3),subt(p1,s1))*dmult(pvec(s1,s2,s3),subt(p2,s1))>eps;
}
//判兩點在平面異側,點在平面上返回0
bool opposite_side(point3 p1,point3 p2,plane3 s)
{
return dmult(pvec(s),subt(p1,s.a))*dmult(pvec(s),subt(p2,s.a))<-eps;
}
bool opposite_side(point3 p1,point3 p2,point3 s1,point3 s2,point3 s3)
{
return dmult(pvec(s1,s2,s3),subt(p1,s1))*dmult(pvec(s1,s2,s3),subt(p2,s1))<-eps;
}
//判兩直線平行
bool parallel(line3 u,line3 v)
{
return vlen(xmult(subt(u.a,u.b),subt(v.a,v.b)))<eps;
}
//判兩平面平行
bool parallel(plane3 u,plane3 v)
{
return vlen(xmult(pvec(u),pvec(v)))<eps;
}
//判直線與平面平行
bool parallel(line3 l,plane3 s)
{
return zero(dmult(subt(l.a,l.b),pvec(s)));
}
bool parallel(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3)
{
return zero(dmult(subt(l1,l2),pvec(s1,s2,s3)));
}
//判兩直線垂直
bool perpendicular(line3 u,line3 v)
{
return zero(dmult(subt(u.a,u.b),subt(v.a,v.b)));
}
//判兩平面垂直
bool perpendicular(plane3 u,plane3 v)
{
return zero(dmult(pvec(u),pvec(v)));
}
//判直線與平面平行
bool perpendicular(line3 l,plane3 s)
{
return vlen(xmult(subt(l.a,l.b),pvec(s)))<eps;
}
//判兩線段相交,包括端點和部分重合
bool intersect_in(line3 u,line3 v)
{
if (!dots_onplane(u.a,u.b,v.a,v.b))
return 0;
if (!dots_inline(u.a,u.b,v.a)||!dots_inline(u.a,u.b,v.b))
return !same_side(u.a,u.b,v)&&!same_side(v.a,v.b,u);
return dot_online_in(u.a,v)||dot_online_in(u.b,v)||dot_online_in(v.a,u)||dot_online_in(v.b,u);
}
//判兩線段相交,不包括端點和部分重合
bool intersect_ex(line3 u,line3 v)
{
return dots_onplane(u.a,u.b,v.a,v.b)&&opposite_side(u.a,u.b,v)&&opposite_side(v.a,v.b,u);
}
//判線段與空間三角形相交,包括交於邊界和(部分)包含
bool intersect_in(line3 l,plane3 s)
{
return !same_side(l.a,l.b,s)&&!same_side(s.a,s.b,l.a,l.b,s.c)&&!same_side(s.b,s.c,l.a,l.b,s.a)&&!same_side(s.c,s.a,l.a,l.b,s.b);
}
//判線段與空間三角形相交,不包括交於邊界和(部分)包含
bool intersect_ex(line3 l,plane3 s)
{
return opposite_side(l.a,l.b,s)&&opposite_side(s.a,s.b,l.a,l.b,s.c)&&opposite_side(s.b,s.c,l.a,l.b,s.a)&&opposite_side(s.c,s.a,l.a,l.b,s.b);
}
//計算兩直線交點,注意事先判斷直線是否共面和平行!
//線段交點請另外判線段相交(同時還是要判斷是否平行!)
point3 intersection(line3 u,line3 v)
{
point3 ret=u.a;
double t=((u.a.x-v.a.x)*(v.a.y-v.b.y)-(u.a.y-v.a.y)*(v.a.x-v.b.x))
/((u.a.x-u.b.x)*(v.a.y-v.b.y)-(u.a.y-u.b.y)*(v.a.x-v.b.x));
ret.x+=(u.b.x-u.a.x)*t;
ret.y+=(u.b.y-u.a.y)*t;
ret.z+=(u.b.z-u.a.z)*t;
return ret;
}
//計算直線與平面交點,注意事先判斷是否平行,並保證三點不共線!
//線段和空間三角形交點請另外判斷
point3 intersection(line3 l,plane3 s)
{
point3 ret=pvec(s);
double t=(ret.x*(s.a.x-l.a.x)+ret.y*(s.a.y-l.a.y)+ret.z*(s.a.z-l.a.z))/(ret.x*(l.b.x-l.a.x)+ret.y*(l.b.y-l.a.y)+ret.z*(l.b.z-l.a.z));
ret.x=l.a.x+(l.b.x-l.a.x)*t;
ret.y=l.a.y+(l.b.y-l.a.y)*t;
ret.z=l.a.z+(l.b.z-l.a.z)*t;
return ret;
}
point3 intersection(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3)
{
point3 ret=pvec(s1,s2,s3);
double t=(ret.x*(s1.x-l1.x)+ret.y*(s1.y-l1.y)+ret.z*(s1.z-l1.z))/
(ret.x*(l2.x-l1.x)+ret.y*(l2.y-l1.y)+ret.z*(l2.z-l1.z));
ret.x=l1.x+(l2.x-l1.x)*t;
ret.y=l1.y+(l2.y-l1.y)*t;
ret.z=l1.z+(l2.z-l1.z)*t;
return ret;
}
//計算兩平面交線,注意事先判斷是否平行,並保證三點不共線!
line3 intersection(plane3 u,plane3 v)
{
line3 ret;
ret.a=parallel(v.a,v.b,u.a,u.b,u.c)?intersection(v.b,v.c,u.a,u.b,u.c):intersection(v.a,v.b,u.a,u.b,u.c);
ret.b=parallel(v.c,v.a,u.a,u.b,u.c)?intersection(v.b,v.c,u.a,u.b,u.c):intersection(v.c,v.a,u.a,u.b,u.c);
return ret;
}
line3 intersection(point3 u1,point3 u2,point3 u3,point3 v1,point3 v2,point3 v3)
{
line3 ret;
ret.a=parallel(v1,v2,u1,u2,u3)?intersection(v2,v3,u1,u2,u3):intersection(v1,v2,u1,u2,u3);
ret.b=parallel(v3,v1,u1,u2,u3)?intersection(v2,v3,u1,u2,u3):intersection(v3,v1,u1,u2,u3);
return ret;
}
//點到直線距離
double ptoline(point3 p,line3 l)
{
return vlen(xmult(subt(p,l.a),subt(l.b,l.a)))/distance(l.a,l.b);
}
//點到平面距離
double ptoplane(point3 p,plane3 s)
{
return fabs(dmult(pvec(s),subt(p,s.a)))/vlen(pvec(s));
}
//直線到直線距離
double linetoline(line3 u,line3 v)
{
point3 n=xmult(subt(u.a,u.b),subt(v.a,v.b));
return fabs(dmult(subt(u.a,v.a),n))/vlen(n);
}
//兩直線夾角cos 值
double angle_cos(line3 u,line3 v)
{
return dmult(subt(u.a,u.b),subt(v.a,v.b))/vlen(subt(u.a,u.b))/vlen(subt(v.a,v.b));
}
//兩平面夾角cos 值
double angle_cos(plane3 u,plane3 v)
{
return dmult(pvec(u),pvec(v))/vlen(pvec(u))/vlen(pvec(v));
}
//直線平面夾角sin 值
double angle_sin(line3 l,plane3 s)
{
return dmult(subt(l.a,l.b),pvec(s))/vlen(subt(l.a,l.b))/vlen(pvec(s));
}
int main()
{
return 0;
}
