LA2572 Viva Confetti 计算几何
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给若干个圆,按顺序从下往上放,全部放完后由几个圆是可见的。
对每个圆,枚举其他圆,记录下所有相交产生的圆弧,然后枚举圆弧(这里可以用圆弧上的任一点来表示圆弧,比如中点)找到最靠上的并且包含这段圆弧的圆,那么这个圆一定是可见的。总复杂度O(n^3),(对每个圆,最多会产生n段相交的圆弧)。
#include <iostream>#include <cstdio>#include <algorithm>#include <cstring>#include <cmath>#include <string>#include <vector>typedef double type;using namespace std;const double PI=acos(-1.0);const double eps=1e-13;struct Point{ type x,y; Point(){} Point(type a,type b) { x=a; y=b; } void read() { scanf("%lf%lf",&x,&y); } void print() { printf("%.6lf %.6lf",x,y); }};typedef Point Vector;Vector operator + (Vector A,Vector B){ return Vector(A.x+B.x,A.y+B.y);}Vector operator - (Point A,Point B){ return Vector(A.x-B.x,A.y-B.y);}Vector operator * (Vector A,type p){ return Vector(A.x*p,A.y*p);}Vector operator / (Vector A,type p){ return Vector(A.x/p,A.y/p);}bool operator < (const Point &a,const Point &b){ return a.x<b.x || (a.x==b.x && a.y<b.y);}int dcmp(double x){ if (fabs(x)<eps) return 0; else return x<0?-1:1;}bool operator == (const Point& a,const Point b){ return dcmp(a.x-b.x)==0 && dcmp(a.y-b.y)==0;}//atan2(x,y) :向量(x,y)的极角,即从x轴正半轴旋转到该向量方向所需要的角度。type Dot(Vector A,Vector B){ return A.x*B.x+A.y*B.y;}type Cross(Vector A,Vector B){ return A.x*B.y-A.y*B.x;}type Length(Vector A){ return sqrt(Dot(A,A));}type Angle(Vector A,Vector B){ return acos(Dot(A,B))/Length(A)/Length(B);}type Area2(Point A,Point B,Point C){ return Cross(B-A,C-A);}Vector Rotate(Vector A,double rad){ return Vector(A.x*cos(rad)-A.y*sin(rad),A.x*sin(rad)+A.y*cos(rad));}Vector Normal(Vector A)//单位法线,左转90度,长度归一{ double L=Length(A); return Vector(-A.y/L,A.x/L);}Point GetLineIntersection(Point P,Vector v,Point Q,Vector w){ Vector u=P-Q; double t=Cross(w,u)/Cross(v,w); return P+v*t;}double DistanceToLine(Point P,Point A,Point B){ Vector v1=B-A,v2=P-A; return fabs(Cross(v1,v2))/Length(v1);}double DistanceToSegment(Point P,Point A,Point B){ if (A==B) return Length(P-A); Vector v1=B-A,v2=P-A,v3=P-B; if (dcmp(Dot(v1,v2))<0) return Length(v2); else if (dcmp(Dot(v1,v3))>0) return Length(v3); else return fabs(Cross(v1,v2))/Length(v1);}Point GetLineProjection(Point P,Point A,Point B)//P在AB上的投影{ Vector v=B-A; return A+v*(Dot(v,P-A)/Dot(v,v));}bool SegmentProperIntersection(Point a1,Point a2,Point b1,Point b2){ double c1=Cross(a2-a1,b1-a1),c2=Cross(a2-a1,b2-a1), c3=Cross(b2-b1,a1-b1),c4=Cross(b2-b1,a2-b1); return dcmp(c1)*dcmp(c2)<0 && dcmp(c3)*dcmp(c4)<0;}bool OnSegment(Point p,Point a1,Point a2){ return dcmp(Cross(a1-p,a2-p))==0 && dcmp(Dot(a1-p,a2-p))<0;}double ConvexPolygonArea(Point* p,int n)//多边形面积{ double area=0; for (int i=1; i<n-1; i++) area+=Cross(p[i]-p[0],p[i+1]-p[0]); return area/2.0;}double PolygonArea(Point* p,int n)//有向面积{ double area=0; for (int i=1; i<n-1; i++) area+=Cross(p[i]-p[0],p[i+1]-p[0]); return area/2.0;}struct Line{ Point p; Vector v; double ang; Line(){}; Line(Point PP,Vector vv) { p=PP; v=vv; ang=atan2(v.y,v.x); } bool operator< (const Line& L)const { return ang<L.ang; } Point point(double t) { return p+v*t; }};struct Circle{ Point c; double r; Circle() { } Circle(Point cc,double rr) { c=cc; r=rr; } Point point(double a) { return Point(c.x+cos(a)*r,c.y+sin(a)*r); }};int getLineCircleIntersection(Line L,Circle C,double& t1,double &t2,vector<Point>& sol){ double a=L.v.x, b=L.p.x-C.c.x, c=L.v.y, d=L.p.y-C.c.y; double e=a*a+c*c,f=2*(a*b+c*d), g=b*b+d*d-C.r*C.r; double delta=f*f-4*e*g;//判别式 if (dcmp(delta)<0) return 0;//相离 if (dcmp(delta)==0) { t1=t2=-f/(2*e); sol.push_back(L.point(t1)); return 1; }//相切 t1=(-f-sqrt(delta))/(2*e); sol.push_back(L.point(t1)); t2=(-f+sqrt(delta))/(2*e); sol.push_back(L.point(t2)); return 2;}double angle(Vector v)//向量极角{ return atan2(v.y,v.x);}int getCircleCircleIntersection(Circle C1,Circle C2,vector<Point>& sol){ double d = Length(C1.c-C2.c); if (dcmp(d)==0) { if (dcmp(C1.r-C2.r)==0) return -1; return 0; } if (dcmp(C1.r+C2.r-d)<0) return 0; if (dcmp(fabs(C1.r-C2.r)-d)>0) return 0; double a=angle(C2.c-C1.c); double da= acos((C1.r*C1.r+d*d-C2.r*C2.r)/(2*C1.r*d)); Point p1=C1.point(a-da),p2=C1.point(a+da); sol.push_back(p1); if (p1==p2) return 1; sol.push_back(p2); return 2;}int getTangents(Point p,Circle C,Vector* v){ Vector u=C.c-p; double dist=Length(u); if (dist<C.r) return 0; else if (dcmp(dist-C.r)==0) { v[0]=Rotate(u,PI/2); return 1; } else { double ang=asin(C.r/dist); v[0]=Rotate(u,-ang); v[1]=Rotate(u,+ang); return 2; }}int getTangents(Circle A,Circle B,Point* a,Point* b){ int cnt=0; if (A.r<B.r) { swap(A,B); swap(a,b); } int d2=(A.c.x-B.c.x)*(A.c.x-B.c.x)+(A.c.y-B.c.y)*(A.c.y-B.c.y); int rdiff=A.r-B.r; int rsum=A.r+B.r; if (d2<rdiff*rdiff) return 0; double base=atan2(B.c.y-A.c.y,B.c.x-A.c.x); if (d2==0 && A.r==B.r) return -1; if (d2==rdiff*rdiff) { a[cnt]=A.point(base); b[cnt]=B.point(base); cnt++; return 1; } double ang=acos((A.r-B.r)/sqrt(d2)); a[cnt]=A.point(base+ang); b[cnt]=B.point(base+ang); cnt++; a[cnt]=A.point(base-ang); b[cnt]=B.point(base-ang); cnt++; if (d2==rsum*rsum) { a[cnt]=A.point(base); b[cnt]=B.point(PI+base); cnt++; } else if (d2>rsum*rsum) { double ang=acos((A.r+B.r)/sqrt(d2)); a[cnt]=A.point(base+ang); b[cnt]=B.point(PI+base+ang); cnt++; a[cnt]=A.point(base-ang); b[cnt]=B.point(PI+base-ang); cnt++; } return cnt;}void cg(double &a){// a=a*180/PI; if (a<0) a+=PI*2; if (a>=PI*2) a-=PI*2;}double ang[420];int cnt;bool ok[120];int n;int main(){// freopen("in.txt","r",stdin); Circle cir[110]; while(~scanf("%d",&n) && n) { memset(ok,false,sizeof ok); for (int i=0; i<n; i++) cir[i].c.read(),scanf("%lf",&cir[i].r); for (int i=0; i<n; i++) { memset(ang,0,sizeof ang); cnt=0; ang[cnt++]=0.0; ang[cnt++]=PI*2.0; for (int j=0; j<n; j++) { if (i==j || Length(cir[i].c-cir[j].c)>cir[i].r+cir[j].r || Length(cir[i].c-cir[j].c)<cir[i].r-cir[j].r) continue; double ds=Length(cir[i].c-cir[j].c); double an=atan2(cir[j].c.y-cir[i].c.y,cir[j].c.x-cir[i].c.x); double ant=acos((cir[i].r*cir[i].r+ds*ds-cir[j].r*cir[j].r)/2.0/cir[i].r/ds); double an1=an+ant,an2=an-ant; cg(an1); cg(an2); ang[cnt++]=an1; ang[cnt++]=an2; } sort(ang,ang+cnt); for (int j=1; j<cnt; j++) { double an=((ang[j]+ang[j-1])/2); for (int tp=-1; tp<=1; tp+=2) { double rr=cir[i].r+tp*eps; Point p=Point(cir[i].c.x+rr*cos(an),cir[i].c.y+rr*sin(an)); for (int k=n-1; k>=0; k--) { if (Length(cir[k].c-p)<cir[k].r) { ok[k]=true; break; } } } } } int ans=0; for (int i=0; i<n; i++) if (ok[i]) ans++; cout<<ans<<endl; } return 0;}
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