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#include "Geometry/FracCoord.hpp"
#pragma once #include "Math/fraction.hpp" struct Point{ Frac X,Y; Point():X(0),Y(0){} Point(Frac _X,Frac _Y):X(_X),Y(_Y){} int pos()const{ if(Y<0)return -1; if(Y==0 and X>=0)return 0; return 1; } Point& operator+=(const Point& p){ X+=p.X; Y+=p.Y; return *this; } Point& operator-=(const Point& p){ X-=p.X; Y-=p.Y; return *this; } Point& operator*=(const Frac& t){ X*=t,Y*=t; return *this; } Point& operator*=(const Point& p){ Frac NX=X*p.X-Y*p.Y,NY=X*p.Y+Y*p.X; X=NX,Y=NY; return *this; } Point operator+(const Point &p) const { return Point(*this) += p; } Point operator-(const Point &p) const { return Point(*this) -= p; } Point operator*(const Frac &p) const { return Point(*this) *= p; } Point operator*(const Point &p) const { return Point(*this) *= p; } Point operator-() const { return Point(-X, -Y); } bool operator==(const Point &p) const { return X == p.X && Y == p.Y; } bool operator!=(const Point &p) const { return X != p.X || Y != p.Y; } bool operator<(const Point &p) const { return X == p.X ? Y < p.Y : X < p.X; } }; struct Line{ Point a,b; Line(){} Line(Point _a,Point _b):a(_a),b(_b){} }; struct Segment:Line{ Segment(){} Segment(Point _a,Point _b):Line(_a,_b){} }; using Poly=vector<Point>; Frac dot(const Point &a, const Point &b) { return a.X * b.X + a.Y * b.Y; } Frac cross(const Point &a, const Point &b) { return a.X * b.Y - a.Y * b.X; } Frac norm(const Point& a){return a.X*a.X+a.Y*a.Y;} bool cmp(const Point& a,const Point& b){ if(a.pos()!=b.pos())return a.pos()<b.pos(); return a.Y*b.X<a.X*b.Y; } Point Projection(const Line&l,const Point& p){ Frac t=dot(p-l.a,l.a-l.b)/norm(l.a-l.b); return l.a+(l.a-l.b)*t; } Point Projection(const Segment&l,const Point& p){ Frac t=dot(p-l.a,l.a-l.b)/norm(l.a-l.b); return l.a+(l.a-l.b)*t; } Point Reflection(const Line& l,const Point& p){ return p+(Projection(l,p)-p)*2.; } int ccw(const Point& a,Point b,Point c){ b-=a; c-=a; Frac C=cross(b,c); if(C>0)return 1; //ccw if(C<0)return -1; //cw if(dot(b,c)<0)return 2; //c,a,b if(dot(b,b)<dot(c,c))return -2; //a,b,c return 0; //a,c,b } bool isOrthogonal(const Line& a,const Line& b){ return dot(a.b-a.a,b.b-b.a)==0; } bool isParallel(const Line& a,const Line& b){ return cross(a.b-a.a,b.b-b.a)==0; } bool isIntersect(const Segment& a,const Segment& b){ return ccw(a.a,a.b,b.a)*ccw(a.a,a.b,b.b)<=0 and ccw(b.a,b.b,a.a)*ccw(b.a,b.b,a.b)<=0; } Point Intersection(const Line& a,const Line& b){ Frac d1=cross(a.b-a.a,b.b-b.a); Frac d2=cross(a.b-a.a,a.b-b.a); if(d1==0 and d2==0)return b.a; return b.a+(b.b-b.a)*(d2/d1); } Frac Area(const Poly& a){ Frac res=0; int n=a.size(); rep(i,0,n)res+=cross(a[i],a[(i+1)%n]); return res/2; } int isContained(const Poly& a,const Point& b){ // 0:not contain,1:on edge,2:contain bool res=0; int n=a.size(); rep(i,0,n){ Point p=a[i]-b,q=a[(i+1)%n]-b; if(p.Y>q.Y)swap(p,q); if(p.Y<=0 and q.Y>0 and cross(p,q)>0)res^=1; if(cross(p,q)==0 and dot(p,q)<=0)return 1; } return (res?2:0); } Poly ConvexHull(Poly& a){ int n=a.size(),k=0; if(n<=2)return a; sort(ALL(a),[](const Point& p,const Point& q){ return (p.Y==q.Y?p.X<q.X:p.Y<q.Y); }); Poly res(n*2); for(int i=0;i<n;res[k++]=a[i++]){ while(k>=2 and cross(res[k-1]-res[k-2],a[i]-res[k-1])<0)k--; } for(int i=n-2,t=k+1;i>=0;res[k++]=a[i--]){ while(k>=t and cross(res[k-1]-res[k-2],a[i]-res[k-1])<0)k--; } res.resize(k-1); return res; } Poly Cut(const Poly& a,const Line& l){ int n=a.size(); Poly res; rep(i,0,n){ Point p=a[i],q=a[(i+1)%n]; if(ccw(l.a,l.b,p)!=-1)res.push_back(p); if(ccw(l.a,l.b,p)*ccw(l.a,l.b,q)<0)res.push_back(Intersection(Line(p,q),l)); } return res; } /** * @brief Geometry(Fraction Coordinates) */
#line 2 "Math/fraction.hpp" template <typename T> struct Frac { T a, b; Frac(T _a = 0) { init(_a, 1); } Frac(T _a, T _b) { init(_a, _b); } Frac &init(T _a, T _b) { T g = GCD(_a, _b); a = _a / g, b = _b / g; if (b < 0) a = -a, b = -b; return *this; } Frac inv() const { return Frac(b, a); } Frac operator-() const { return Frac(-a, b); } Frac &operator+=(const Frac &x) { return init(a * x.b + x.a * b, b * x.b); } Frac &operator-=(const Frac &x) { return init(a * x.b - x.a * b, b * x.b); } Frac &operator*=(const Frac &x) { return init(a * x.a, b * x.b); } Frac &operator/=(const Frac &x) { return init(a * x.b, b * x.a); } Frac operator+(const Frac &x) const { return Frac(*this) += x; } Frac operator-(const Frac &x) const { return Frac(*this) -= x; } Frac operator*(const Frac &x) const { return Frac(*this) *= x; } Frac operator/(const Frac &x) const { return Frac(*this) /= x; } bool operator<(const Frac &x) const { return a * x.b < b * x.a; } bool operator>(const Frac &x) const { return x < *this; } bool operator<=(const Frac &x) const { return !(x < *this); } bool operator>=(const Frac &x) const { return !(*this < x); } bool operator==(const Frac &x) const { return (*this <= x and x <= *this); } bool operator!=(const Frac &x) const { return !(*this == x); } T GCD(T a, T b) { if (b == 0) return a; else return GCD(b, a % b); } }; template <typename T> Frac<T> between(const Frac<T> &x, const Frac<T> &y) { if (x.a < x.b and y.b < y.a) return Frac(1); else if (x.b <= x.a) { T add = floor(x.a / x.b); return between(x - add, y - add) + add; } else return between(y.inv(), x.inv()).inv(); } /** * @brief Fraction * @docs docs/fraction.md */ #line 3 "Geometry/FracCoord.hpp" struct Point{ Frac X,Y; Point():X(0),Y(0){} Point(Frac _X,Frac _Y):X(_X),Y(_Y){} int pos()const{ if(Y<0)return -1; if(Y==0 and X>=0)return 0; return 1; } Point& operator+=(const Point& p){ X+=p.X; Y+=p.Y; return *this; } Point& operator-=(const Point& p){ X-=p.X; Y-=p.Y; return *this; } Point& operator*=(const Frac& t){ X*=t,Y*=t; return *this; } Point& operator*=(const Point& p){ Frac NX=X*p.X-Y*p.Y,NY=X*p.Y+Y*p.X; X=NX,Y=NY; return *this; } Point operator+(const Point &p) const { return Point(*this) += p; } Point operator-(const Point &p) const { return Point(*this) -= p; } Point operator*(const Frac &p) const { return Point(*this) *= p; } Point operator*(const Point &p) const { return Point(*this) *= p; } Point operator-() const { return Point(-X, -Y); } bool operator==(const Point &p) const { return X == p.X && Y == p.Y; } bool operator!=(const Point &p) const { return X != p.X || Y != p.Y; } bool operator<(const Point &p) const { return X == p.X ? Y < p.Y : X < p.X; } }; struct Line{ Point a,b; Line(){} Line(Point _a,Point _b):a(_a),b(_b){} }; struct Segment:Line{ Segment(){} Segment(Point _a,Point _b):Line(_a,_b){} }; using Poly=vector<Point>; Frac dot(const Point &a, const Point &b) { return a.X * b.X + a.Y * b.Y; } Frac cross(const Point &a, const Point &b) { return a.X * b.Y - a.Y * b.X; } Frac norm(const Point& a){return a.X*a.X+a.Y*a.Y;} bool cmp(const Point& a,const Point& b){ if(a.pos()!=b.pos())return a.pos()<b.pos(); return a.Y*b.X<a.X*b.Y; } Point Projection(const Line&l,const Point& p){ Frac t=dot(p-l.a,l.a-l.b)/norm(l.a-l.b); return l.a+(l.a-l.b)*t; } Point Projection(const Segment&l,const Point& p){ Frac t=dot(p-l.a,l.a-l.b)/norm(l.a-l.b); return l.a+(l.a-l.b)*t; } Point Reflection(const Line& l,const Point& p){ return p+(Projection(l,p)-p)*2.; } int ccw(const Point& a,Point b,Point c){ b-=a; c-=a; Frac C=cross(b,c); if(C>0)return 1; //ccw if(C<0)return -1; //cw if(dot(b,c)<0)return 2; //c,a,b if(dot(b,b)<dot(c,c))return -2; //a,b,c return 0; //a,c,b } bool isOrthogonal(const Line& a,const Line& b){ return dot(a.b-a.a,b.b-b.a)==0; } bool isParallel(const Line& a,const Line& b){ return cross(a.b-a.a,b.b-b.a)==0; } bool isIntersect(const Segment& a,const Segment& b){ return ccw(a.a,a.b,b.a)*ccw(a.a,a.b,b.b)<=0 and ccw(b.a,b.b,a.a)*ccw(b.a,b.b,a.b)<=0; } Point Intersection(const Line& a,const Line& b){ Frac d1=cross(a.b-a.a,b.b-b.a); Frac d2=cross(a.b-a.a,a.b-b.a); if(d1==0 and d2==0)return b.a; return b.a+(b.b-b.a)*(d2/d1); } Frac Area(const Poly& a){ Frac res=0; int n=a.size(); rep(i,0,n)res+=cross(a[i],a[(i+1)%n]); return res/2; } int isContained(const Poly& a,const Point& b){ // 0:not contain,1:on edge,2:contain bool res=0; int n=a.size(); rep(i,0,n){ Point p=a[i]-b,q=a[(i+1)%n]-b; if(p.Y>q.Y)swap(p,q); if(p.Y<=0 and q.Y>0 and cross(p,q)>0)res^=1; if(cross(p,q)==0 and dot(p,q)<=0)return 1; } return (res?2:0); } Poly ConvexHull(Poly& a){ int n=a.size(),k=0; if(n<=2)return a; sort(ALL(a),[](const Point& p,const Point& q){ return (p.Y==q.Y?p.X<q.X:p.Y<q.Y); }); Poly res(n*2); for(int i=0;i<n;res[k++]=a[i++]){ while(k>=2 and cross(res[k-1]-res[k-2],a[i]-res[k-1])<0)k--; } for(int i=n-2,t=k+1;i>=0;res[k++]=a[i--]){ while(k>=t and cross(res[k-1]-res[k-2],a[i]-res[k-1])<0)k--; } res.resize(k-1); return res; } Poly Cut(const Poly& a,const Line& l){ int n=a.size(); Poly res; rep(i,0,n){ Point p=a[i],q=a[(i+1)%n]; if(ccw(l.a,l.b,p)!=-1)res.push_back(p); if(ccw(l.a,l.b,p)*ccw(l.a,l.b,q)<0)res.push_back(Intersection(Line(p,q),l)); } return res; } /** * @brief Geometry(Fraction Coordinates) */