Computational Geometry Template_Fundamental Rules

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#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;
};


//计算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);
}
//计算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);
}
//计算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;
}
//计算dot product U . V
double dmult(point3 u,point3 v)
{
    return u.x*v.x+u.y*v.y+u.z*v.z;
}


//两点距离
double distance(point p1,point p2)
{
    return sqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));
}

//判三点共线
bool dots_inline(point p1,point p2,point p3)
{
    return zero(xmult(p1,p2,p3));
}

//判点是否在线段上,包括端点
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;
}

//判点是否在线段上,不包括端点
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));
}

//判两点在线段同侧,点在线段上返回0
bool same_side(point p1,point p2,line l)
{
    return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)>eps;
}

//判两点在线段异侧,点在线段上返回0
bool opposite_side(point p1,point p2,line l)
{
    return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)<-eps;
}

//判两直线平行
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));
}

//判两直线垂直
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));
}

//判两线段相交,包括端点和部分重合
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);
}

//判两线段相交,不包括端点和部分重合
bool intersect_ex(line u,line v)
{
    return opposite_side(u.a,u.b,v)&&opposite_side(v.a,v.b,u);
}

//计算两直线交点,注意事先判断直线是否平行!
//线段交点请另外判线段相交(同时还是要判断是否平行!)
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;
}
//点到直线上的最近点
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);
}

//点到直线距离
double disptoline(point p,line l)
{
    return fabs(xmult(p,l.a,l.b))/distance(l.a,l.b);
}

//点到线段上的最近点
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);
}

//点到线段距离
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 为顶点逆时针旋转angle 并放大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;
}
//计算三角形面积,输入三顶点
double area_triangle(point p1,point p2,point p3)
{
    return fabs(xmult(p1,p2,p3))/2;
}

//计算三角形面积,输入三边长
double area_triangle(double a,double b,double c)
{
    double s=(a+b+c)/2;
    return sqrt(s*(s-a)*(s-b)*(s-c));
}
//计算多边形面积,顶点按顺时针或逆时针给出
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;
}

//计算圆心角lat 表示纬度,-90<=w<=90,lng 表示经度
//返回两点所在大圆劣弧对应圆心角,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));
}
//计算距离,r 为球半径
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)));
}
//计算球面距离,r 为球半径
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);
}
//内心
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
//到三角形三顶点距离的平方和最小的点
//三角形内到三边距离之积最大的点
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);
}
//费马点
//到三角形三顶点距离之和最小的点
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));
}
//两点距离,单参数取向量大小
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));
}


///三维///
//向量大小
double vlen(point3 p)
{
    return sqrt(p.x*p.x+p.y*p.y+p.z*p.z);
}

//判三点共线
bool dots_inline(point3 p1,point3 p2,point3 p3)
{
    return vlen(xmult(subt(p1,p2),subt(p2,p3)))<eps;
}

//判四点共面
bool dots_onplane(point3 a,point3 b,point3 c,point3 d)
{
    return zero(dmult(pvec(a,b,c),subt(d,a)));
}

//判点是否在线段上,包括端点和共线
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;
}

//判点是否在线段上,不包括端点
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));
}

//判点是否在空间三角形上,包括边界,三点共线无意义
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))));
}

//判点是否在空间三角形上,不包括边界,三点共线无意义
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;
}

//判两点在线段同侧,点在线段上返回0,不共面无意义
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;
}

//判两点在线段异侧,点在线段上返回0,不共面无意义
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;
}

//判两点在平面同侧,点在平面上返回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;
}

//判两点在平面异侧,点在平面上返回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;
}
//判两直线平行
bool parallel(line3 u,line3 v)
{
    return vlen(xmult(subt(u.a,u.b),subt(v.a,v.b)))<eps;
}

//判两平面平行
bool parallel(plane3 u,plane3 v)
{
    return vlen(xmult(pvec(u),pvec(v)))<eps;
}

//判直线与平面平行
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)));
}

//判两直线垂直
bool perpendicular(line3 u,line3 v)
{
    return zero(dmult(subt(u.a,u.b),subt(v.a,v.b)));
}

//判两平面垂直
bool perpendicular(plane3 u,plane3 v)
{
    return zero(dmult(pvec(u),pvec(v)));
}

//判直线与平面平行
bool perpendicular(line3 l,plane3 s)
{
    return vlen(xmult(subt(l.a,l.b),pvec(s)))<eps;
}

//判两线段相交,包括端点和部分重合
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);
}

//判两线段相交,不包括端点和部分重合
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);
}

//判线段与空间三角形相交,包括交于边界和(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);
}

//判线段与空间三角形相交,不包括交于边界和(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);
}

//计算两直线交点,注意事先判断直线是否共面和平行!
//线段交点请另外判线段相交(同时还是要判断是否平行!)
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;
}

//计算直线与平面交点,注意事先判断是否平行,并保证三点不共线!
//线段和空间三角形交点请另外判断
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;
}

//计算两平面交线,注意事先判断是否平行,并保证三点不共线!
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;
}
//点到直线距离
double ptoline(point3 p,line3 l)
{
    return vlen(xmult(subt(p,l.a),subt(l.b,l.a)))/distance(l.a,l.b);
}

//点到平面距离
double ptoplane(point3 p,plane3 s)
{
    return fabs(dmult(pvec(s),subt(p,s.a)))/vlen(pvec(s));
}

//直线到直线距离
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);
}

//两直线夹角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));
}

//两平面夹角cos 值
double angle_cos(plane3 u,plane3 v)
{
    return dmult(pvec(u),pvec(v))/vlen(pvec(u))/vlen(pvec(v));
}

//直线平面夹角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;
}