//三維幾何函數庫
#include <math.h>
#define eps 1e-8
#define zero(x) (((x)>0?(x):-(x))<eps)
struct point3{double x,y,z;};
struct line3{point3 a,b;};
struct plane3{point3 a,b,c;};

//計算cross product U x 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;
}

//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));
}

//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
int dots_inline(point3 p1,point3 p2,point3 p3){
	return vlen(xmult(subt(p1,p2),subt(p2,p3)))<eps;
}

//判四點共面
int 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
int 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;
}
int dot_online_in(point3 p,point3 l1,point3 l2){
	return zero(vlen(xmult(subt(p,l1),subt(p,l2))))&&(l1.x-p.x)*(l2.x-p.x)<eps&&
		(l1.y-p.y)*(l2.y-p.y)<eps&&(l1.z-p.z)*(l2.z-p.z)<eps;
}

//判點是否在線段上,不包括端點
int 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));
}
int dot_online_ex(point3 p,point3 l1,point3 l2){
	return dot_online_in(p,l1,l2)&&(!zero(p.x-l1.x)||!zero(p.y-l1.y)||!zero(p.z-l1.z))&&
		(!zero(p.x-l2.x)||!zero(p.y-l2.y)||!zero(p.z-l2.z));
}

//判點是否在空間三角形上,包括邊界,三點共線無意義
int 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))));
}
int dot_inplane_in(point3 p,point3 s1,point3 s2,point3 s3){
	return zero(vlen(xmult(subt(s1,s2),subt(s1,s3)))-vlen(xmult(subt(p,s1),subt(p,s2)))-
		vlen(xmult(subt(p,s2),subt(p,s3)))-vlen(xmult(subt(p,s3),subt(p,s1))));
}

//判點是否在空間三角形上,不包括邊界,三點共線無意義
int 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;
}
int dot_inplane_ex(point3 p,point3 s1,point3 s2,point3 s3){
	return dot_inplane_in(p,s1,s2,s3)&&vlen(xmult(subt(p,s1),subt(p,s2)))>eps&&
		vlen(xmult(subt(p,s2),subt(p,s3)))>eps&&vlen(xmult(subt(p,s3),subt(p,s1)))>eps;
}

//判兩點在線段同側,點在線段上返回0,不共面無意義
int 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;
}
int same_side(point3 p1,point3 p2,point3 l1,point3 l2){
	return dmult(xmult(subt(l1,l2),subt(p1,l2)),xmult(subt(l1,l2),subt(p2,l2)))>eps;
}

//判兩點在線段異側,點在線段上返回0,不共面無意義
int 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;
}
int opposite_side(point3 p1,point3 p2,point3 l1,point3 l2){
	return dmult(xmult(subt(l1,l2),subt(p1,l2)),xmult(subt(l1,l2),subt(p2,l2)))<-eps;
}

//判兩點在平面同側,點在平面上返回0
int same_side(point3 p1,point3 p2,plane3 s){
	return dmult(pvec(s),subt(p1,s.a))*dmult(pvec(s),subt(p2,s.a))>eps;
}
int 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
int opposite_side(point3 p1,point3 p2,plane3 s){
	return dmult(pvec(s),subt(p1,s.a))*dmult(pvec(s),subt(p2,s.a))<-eps;
}
int 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
int parallel(line3 u,line3 v){
	return vlen(xmult(subt(u.a,u.b),subt(v.a,v.b)))<eps;
}
int parallel(point3 u1,point3 u2,point3 v1,point3 v2){
	return vlen(xmult(subt(u1,u2),subt(v1,v2)))<eps;
}

//判兩平面平行
int parallel(plane3 u,plane3 v){
	return vlen(xmult(pvec(u),pvec(v)))<eps;
}
int parallel(point3 u1,point3 u2,point3 u3,point3 v1,point3 v2,point3 v3){
	return vlen(xmult(pvec(u1,u2,u3),pvec(v1,v2,v3)))<eps;
}

//判直線與平面平行
int parallel(line3 l,plane3 s){
	return zero(dmult(subt(l.a,l.b),pvec(s)));
}
int 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
int perpendicular(line3 u,line3 v){
	return zero(dmult(subt(u.a,u.b),subt(v.a,v.b)));
}
int perpendicular(point3 u1,point3 u2,point3 v1,point3 v2){
	return zero(dmult(subt(u1,u2),subt(v1,v2)));
}

//判兩平面垂直
int perpendicular(plane3 u,plane3 v){
	return zero(dmult(pvec(u),pvec(v)));
}
int perpendicular(point3 u1,point3 u2,point3 u3,point3 v1,point3 v2,point3 v3){
	return zero(dmult(pvec(u1,u2,u3),pvec(v1,v2,v3)));
}

//判直線與平面平行
int perpendicular(line3 l,plane3 s){
	return vlen(xmult(subt(l.a,l.b),pvec(s)))<eps;
}
int perpendicular(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3){
	return vlen(xmult(subt(l1,l2),pvec(s1,s2,s3)))<eps;
}

//判兩線段相交,包括端點和部分重合
int 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);
}
int intersect_in(point3 u1,point3 u2,point3 v1,point3 v2){
	if (!dots_onplane(u1,u2,v1,v2))
		return 0;
	if (!dots_inline(u1,u2,v1)||!dots_inline(u1,u2,v2))
		return !same_side(u1,u2,v1,v2)&&!same_side(v1,v2,u1,u2);
	return dot_online_in(u1,v1,v2)||dot_online_in(u2,v1,v2)||dot_online_in(v1,u1,u2)||dot_online_in(v2,u1,u2);
}

//判兩線段相交,不包括端點和部分重合
int 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);
}
int intersect_ex(point3 u1,point3 u2,point3 v1,point3 v2){
	return dots_onplane(u1,u2,v1,v2)&&opposite_side(u1,u2,v1,v2)&&opposite_side(v1,v2,u1,u2);
}

//判線段與空間三角形相交,包括交于邊界和(部分)包含
int 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);
}
int intersect_in(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3){
	return !same_side(l1,l2,s1,s2,s3)&&!same_side(s1,s2,l1,l2,s3)&&
		!same_side(s2,s3,l1,l2,s1)&&!same_side(s3,s1,l1,l2,s2);
}

//判線段與空間三角形相交,不包括交于邊界和(部分)包含
int 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);
}
int intersect_ex(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3){
	return opposite_side(l1,l2,s1,s2,s3)&&opposite_side(s1,s2,l1,l2,s3)&&
		opposite_side(s2,s3,l1,l2,s1)&&opposite_side(s3,s1,l1,l2,s2);
}

//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;
}
point3 intersection(point3 u1,point3 u2,point3 v1,point3 v2){
	point3 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;
	ret.z+=(u2.z-u1.z)*t;
	return ret;
}

//計算直線與平面交點,注意事先判斷是否平行,并保證三點不共線!
//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,并保證三點不共線!
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);
}
double ptoline(point3 p,point3 l1,point3 l2){
	return vlen(xmult(subt(p,l1),subt(l2,l1)))/distance(l1,l2);
}

//點到平面距離
double ptoplane(point3 p,plane3 s){
	return fabs(dmult(pvec(s),subt(p,s.a)))/vlen(pvec(s));
}
double ptoplane(point3 p,point3 s1,point3 s2,point3 s3){
	return fabs(dmult(pvec(s1,s2,s3),subt(p,s1)))/vlen(pvec(s1,s2,s3));
}

//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);
}
double linetoline(point3 u1,point3 u2,point3 v1,point3 v2){
	point3 n=xmult(subt(u1,u2),subt(v1,v2));
	return fabs(dmult(subt(u1,v1),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));
}
double angle_cos(point3 u1,point3 u2,point3 v1,point3 v2){
	return dmult(subt(u1,u2),subt(v1,v2))/vlen(subt(u1,u2))/vlen(subt(v1,v2));
}

//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));
}
double angle_cos(point3 u1,point3 u2,point3 u3,point3 v1,point3 v2,point3 v3){
	return dmult(pvec(u1,u2,u3),pvec(v1,v2,v3))/vlen(pvec(u1,u2,u3))/vlen(pvec(v1,v2,v3));
}

//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));
}
double angle_sin(point3 l1,point3 l2,point3 s1,point3 s2,point3 s3){
	return dmult(subt(l1,l2),pvec(s1,s2,s3))/vlen(subt(l1,l2))/vlen(pvec(s1,s2,s3));
}