二维几何

点

const double inf=1e20;
const double eps=1e-8;
const double pi=acos(-1.0);
const int maxp=1010;//判断正负
int sgn(double x) {if (fabs(x)<eps) return 0;if (x<0) return -1;else return 1;
}
//平方
inline double sqr(double x) {return x*x;
}struct Point {double x,y;Point() {}Point(double _x, double _y) {x=_x;y=_y;}void input() {scanf("%lf%lf",&x,&y);}void output() {printf("%.2f%.2f\n",x,y);}bool operator == (Point b)const {return sgn(x-b.x)==0 && sgn(y-b.y)==0;}bool operator < (Point b)const {return sgn(x-b.x)==0?sgn(y-b.y)<0:x<b.x;}Point operator -(const Point &b)const {return Point(x-b.x,y-b.y);}//叉积double operator ^(const Point &b)const {return x*b.y-y*b.x;}//点积double operator *(const Point &b)const {return x*b.x+y*b.y;}//返回长度double len() {return hypot(x,y);}//返回长度平方double len2() {return x*x+y*y;}//返回两点间距double distance(Point p) {return hypot(x-p.x,y-p.y);}Point operator +(const Point &b)const {return Point(x+b.x,y+b.y);}Point operator *(const double &k)const {return Point(x*k,y*k);}Point operator /(const double &k)const {return Point(x/k,y/k);}//pa和pb的夹角double rad(Point a,Point b) {Point p=*this;return fabs(atan2(fabs((a-p)^(b-p)),(a-p)*(b-p)));}//化为长度为r的向量Point trunc(double r) {double l=len();if (!sgn(l)) return *this;r/=l;return Point(x*r,y*r);}//逆时针旋转 90 度Point rotleft() {return Point(-y,x);}//顺时针旋转 90 度Point rotright() {return Point(y,-x);}//绕着 p 点逆时针旋转 anglePoint rotate(Point p,double angle) {Point v=(*this)-p;double c=cos(angle),s=sin(angle);return Point(p.x+v.x*c-v.y*s,p.y+v.x*s+v.y*c);}
};

线

struct Line {Point s,e;Line() {}Line(Point _s,Point _e) {s=_s;e=_e;}bool operator ==(Line v) {return (s==v.s) && (e==v.e);}//根据一个点和倾斜角 angle 确定直线,0<=angle<πLine(Point p, double angle) {s=p;if (sgn(angle-pi/2)==0) {e=(s+Point(0,1));} else {e=(s+Point(1,tan(angle)));}}//ax+by+c=0Line(double a,double b,double c) {if(sgn(a)==0) {s=Point(0,-c/b);e=Point(1,-c/b);} else if(sgn(b)==0) {s=Point(-c/a,0);e=Point(-c/a,1);} else {s=Point(0,-c/b);e=Point(1,(-c-a)/b);}}void input() {s.input();e.input();}void adjust() {if(e<s) swap(s,e);	}//求线段长度double length() {return s.distance(e);}//返回直线倾斜角 0<=angle<π double angle() {double k=atan2(e.y-s.y,e.x-s.x);if(sgn(k)<0) k+=pi;if(sgn(k-pi)==0) k-= pi;return k;}//点和直线关系// 1 在左侧// 2 在右侧// 3 在直线上 int relation(Point p) {int c=sgn((p-s)^(e-s));if(c<0) return 1; else if(c>0) return 2; else return 3;	 }//点在线段上的判断bool pointonseg(Point p) {return sgn((p-s)^(e-s))==0 && sgn((p-s)*(p-e))<=0;}//两向量平行 (对应直线平行或重合)bool parallel(Line v) {return sgn((e-s)^(v.e-v.s))==0;}//两线段相交判断//2 规范相交//1 非规范相交//0 不相交int segcrossseg(Line v) {int d1=sgn((e-s)^(v.s-s));int d2=sgn((e-s)^(v.e-s));int d3=sgn((v.e-v.s)^(s-v.s));int d4=sgn((v.e-v.s)^(e-v.s));if((d1^d2)==-2&&(d3^d4)==-2)return 2;return (d1==0 && sgn((v.s-s)*(v.s-e))<=0) ||(d2==0 && sgn((v.e-s)*(v.e-e))<=0) ||(d3==0 && sgn((s-v.s)*(s-v.e))<=0) ||(d4==0 && sgn((e-v.s)*(e-v.e))<=0);}//直线和线段相交判断//-*this line -v seg//2 规范相交//1 非规范相交//0 不相交int linecrossseg(Line v) {int d1=sgn((e-s)^(v.s-s));int d2=sgn((e-s)^(v.e-s));if((d1^d2)==-2) return 2;return (d1==0||d2==0);}//两直线关系//0 平行//1 重合//2 相交int linecrossline(Line v) {if((*this).parallel(v))return v.relation(s)==3;return 2;}//求两直线的交点//要保证两直线不平行或重合Point crosspoint(Line v) {double a1 = (v.e-v.s)^(s-v.s);double a2 = (v.e-v.s)^(e-v.s);return Point((s.x*a2-e.x*a1)/(a2-a1),(s.y*a2-e.y*a1)/(a2-a1));}//点到直线的距离double dispointtoline(Point p) {return fabs((p-s)^(e-s))/length();}//点到线段的距离double dispointtoseg(Point p) {if(sgn((p-s)*(e-s))<0 || sgn((p-e)*(s-e))<0)return min(p.distance(s),p.distance(e));return dispointtoline(p);}//返回线段到线段的距离//前提是两线段不相交，相交距离就是 0 了double dissegtoseg(Line v) {return min(min(dispointtoseg(v.s),dispointtoseg(v.e)),min(v.dispointtoseg(s),v.dispointtoseg(e)));}//返回点 p 在直线上的投影Point lineprog(Point p) {return s + ( ((e-s)*((e-s)*(p-s)))/((e-s).len2()) );}//返回点 p 关于直线的对称点Point symmetrypoint(Point p) {Point q = lineprog(p);return Point(2*q.x-p.x,2*q.y-p.y);}
};

圆

struct circle {Point p; //圆心double r; //半径circle() {}circle(Point _p,double _r) {p=_p;r=_r;}circle(double x,double y,double _r) {p=Point(x,y);r=_r;}//三角形的外接圆//需要 Point 的 + / rotate() 以及 Line 的 crosspoint()//利用两条边的中垂线得到圆心circle(Point a,Point b,Point c) {Line u=Line((a+b)/2,((a+b)/2)+((b-a).rotleft()));Line v=Line((b+c)/2,((b+c)/2)+((c-b).rotleft()));p=u.crosspoint(v);r=p.distance(a);}//三角形的内切圆//参数 bool t 没有作用，只是为了和上面外接圆函数区别circle(Point a,Point b,Point c,bool t) {Line u,v;double m=atan2(b.y-a.y,b.x-a.x),n=atan2(c.y-a.y,c.x-a.x);u.s=a;u.e=u.s+Point(cos((n+m)/2),sin((n+m)/2));v.s=b;m=atan2(a.y-b.y,a.x-b.x),n=atan2(c.y-b.y,c.x-b.x);v.e=v.s+Point(cos((n+m)/2),sin((n+m)/2));p=u.crosspoint(v);r=Line(a,b).dispointtoseg(p);}void input() {p.input();scanf("%lf",&r);}void output() {printf("%.2lf-%.2lf-%.2lf\n",p.x,p.y,r);}bool operator == (circle v) {return (p==v.p) && sgn(r-v.r)==0;}bool operator < (circle v)const {return ((p<v.p) || ((p==v.p) && sgn(r-v.r)<0));}//面积double area() {return pi*r*r;}//周长double circumference() {return 2*pi*r;}//点和圆的关系//0 圆外//1 圆上//2 圆内int relation(Point b) {double dst=b.distance(p);if (sgn(dst-r)<0) return 2;else if (sgn(dst-r)==0) return 1;return 0;}//线段和圆的关系//比较的是圆心到线段的距离和半径的关系int relationseg(Line v) {double dst=v.dispointtoseg(p);if (sgn(dst-r)<0) return 2;else if (sgn(dst-r)==0) return 1;return 0;}//直线和圆的关系//比较的是圆心到直线的距离和半径的关系int relationline(Line v) {double dst=v.dispointtoline(p);if (sgn(dst-r)<0) return 2;else if (sgn(dst-r)==0) return 1;return 0;}//两圆的关系//5 相离//4 外切//3 相交//2 内切//1 内含//需要 Point 的 distanceint relationcircle(circle v) {double d=p.distance(v.p);if (sgn(d-r-v.r)>0) return 5;if (sgn(d-r-v.r)==0) return 4;double l=fabs(r-v.r);if (sgn(d-r-v.r)<0 && sgn(d-l)>0) return 3;if (sgn(d-l)==0) return 2;if (sgn(d-l)<0) return 1;}//求两个圆的交点，返回 0 表示没有交点，返回 1 是一个交点，2 是两个交点//需要 relationcircleint pointcrosscircle(circle v,Point &p1,Point &p2) {int rel=relationcircle(v);if (rel==1 || rel==5)return 0;double d=p.distance(v.p);double l=(d*d+r*r-v.r*v.r)/(2*d);double h=sqrt(r*r-l*l);Point tmp=p+(v.p-p).trunc(l);p1=tmp+((v.p-p).rotleft().trunc(h));p2=tmp+((v.p-p).rotright().trunc(h));if(rel==2 || rel==4)return 1;return 2;}//求直线和圆的交点，返回交点个数int pointcrossline(Line v,Point &p1,Point &p2) {if (!(*this).relationline(v)) return 0;Point a=v.lineprog(p);double d=v.dispointtoline(p);d=sqrt(r*r-d*d);if (sgn(d)==0) {p1=a;p2=a;return 1;}p1=a+(v.e-v.s).trunc(d);p2=a-(v.e-v.s).trunc(d);return 2;}//得到过 a,b 两点，半径为 r1 的两个圆int gercircle(Point a,Point b,double r1,circle &c1,circle &c2) {circle x(a,r1),y(b,r1);int t=x.pointcrosscircle(y,c1.p,c2.p);if (!t) return 0;c1.r=c2.r=r;return t;}//得到与直线 u 相切，过点 q, 半径为 r1 的圆int getcircle(Line u,Point q,double r1,circle &c1,circle &c2) {double dis = u.dispointtoline(q);if (sgn(dis-r1*2)>0) return 0;if (sgn(dis)==0) {c1.p=q+((u.e-u.s).rotleft().trunc(r1));c2.p=q+((u.e-u.s).rotright().trunc(r1));c1.r=c2.r=r1;return 2;}Line u1=Line((u.s+(u.e-u.s).rotleft().trunc(r1)),(u.e+(u.e-u.s).rotleft().trunc(r1)));Line u2=Line((u.s + (u.e-u.s).rotright().trunc(r1)),(u.e+(u.e-u.s).rotright().trunc(r1)));circle cc=circle(q,r1);Point p1,p2;if (!cc.pointcrossline(u1,p1,p2))cc.pointcrossline(u2,p1,p2);c1=circle(p1,r1);if (p1==p2) {c2=c1;return 1;}c2=circle(p2,r1);return 2;}//同时与直线 u,v 相切，半径为 r1 的圆int getcircle (Line u,Line v,double r1,circle &c1,circle &c2,circle &c3,circle &c4) {if(u.parallel(v))return 0;//两直线平行Line u1=Line(u.s+(u.e-u.s).rotleft().trunc(r1),u.e+(u.e-u.s).rotleft().trunc(r1));Line u2=Line(u.s+(u.e-u.s).rotright().trunc(r1),u.e+(u.e-u.s).rotright().trunc(r1));Line v1=Line(v.s+(v.e-v.s).rotleft().trunc(r1),v.e+(v.e-v.s).rotleft().trunc(r1));Line v2=Line(v.s+(v.e-v.s).rotright().trunc(r1),v.e+(v.e-v.s).rotright().trunc(r1));c1.r=c2.r=c3.r=c4.r=r1;c1.p=u1.crosspoint(v1);c2.p=u1.crosspoint(v2);c3.p=u2.crosspoint(v1);c4.p=u2.crosspoint(v2);return 4;}//同时与不相交圆 cx,cy 相切，半径为 r1 的圆int getcircle(circle cx,circle cy,double r1,circle &c1,circle &c2) {circle x(cx.p,r1+cx.r),y(cy.p,r1+cy.r);int t=x.pointcrosscircle(y,c1.p,c2.p);if (!t) return 0;c1.r=c2.r=r1;return t;}//过一点作圆的切线 (先判断点和圆的关系)int tangentline(Point q,Line &u,Line &v) {int x=relation(q);if (x==2) return 0;if (x==1) {u=Line(q,q+(q-p).rotleft());v=u;return 1;}double d=p.distance(q);double l=r*r/d;double h=sqrt(r*r-l*l);u=Line(q,p+((q-p).trunc(l)+(q-p).rotleft().trunc(h)));v=Line(q,p+((q-p).trunc(l)+(q-p).rotright().trunc(h)));return 2;}//求两圆相交的面积double areacircle(circle v) {int rel=relationcircle(v);if (rel>=4) return 0.0;if (rel<=2) return min(area(),v.area());double d=p.distance(v.p);double hf=(r+v.r+d)/2.0;double ss=2*sqrt(hf*(hf-r)*(hf-v.r)*(hf-d));double a1=acos((r*r+d*d-v.r*v.r)/(2.0*r*d));a1=a1*r*r;double a2=acos((v.r*v.r+d*d-r*r)/(2.0*v.r*d));a2=a2*v.r*v.r;return a1+a2-ss;}//求圆和三角形 pab 的相交面积double areatriangle(Point a,Point b) {if (sgn((p-a)^(p-b))==0) return 0.0;Point q[5];int len=0;q[len++]=a;Line l(a,b);Point p1,p2;if (pointcrossline(l,q[1],q[2])==2) {if (sgn((a-q[1])*(b-q[1]))<0) q[len++]=q[1];if (sgn((a-q[2])*(b-q[2]))<0) q[len++]=q[2];}q[len++]=b;if (len==4 && sgn((q[0]-q[1])*(q[2]-q[1]))>0) swap(q[1],q[2]);double res=0;for (int i=0; i<len-1; i++) {if (relation(q[i])==0||relation(q[i+1])==0) {double arg=p.rad(q[i],q[i+1]);res+=r*r*arg/2.0;} else {res+=fabs((q[i]-p)^(q[i+1]-p))/2.0;}}return res;}
};