题目链接：Link
Solution
第一次过计算几何黑题，写篇题解纪念一下。
- 问题一：详见代码
- 问题二：详见代码
- 问题三：很简单的问题，不会去看蓝书
- 问题四：由于半径为r，圆心到直线的距离一定为r，满足该条件的点的轨迹是两条直线。而想要过定点，圆心到该点的距离也一定为r，满足该条件的点的轨迹是一个圆。求出三条合法的轨迹的交点即可。
- 问题五：解法类似，每条直线向两侧平移r，得到的四条直线的交点就是答案。
- 问题六：让每个圆向外膨胀r，新得到的两个圆的交点就是答案。
#include<cstdio>
#include<cstring>
#include<cstdlib>
#include<cmath>
#include<vector>
#include<algorithm>
using std::vector;
using std::swap;
struct Point
{
double x,y;
Point(double x=0,double y=0):x(x),y(y){}
};
typedef Point Vector;
inline Point read_point()
{
Point A;
scanf("%lf%lf",&A.x,&A.y);
return A;
}
inline Vector operator+(const Vector &A,const Vector &B)
{ return Vector(A.x+B.x,A.y+B.y); }
inline Vector operator-(const Point &a,const Point &b)
{ return Vector(a.x-b.x,a.y-b.y); }
inline Vector operator*(const Vector &A,double p)
{ return Vector(A.x*p,A.y*p); }
inline Vector operator/(const Vector &A,double p)
{ return Vector(A.x/p,A.y/p); }
inline bool operator<(const Point &a,const Point &b)
{ return a.x<b.x||(a.x==b.x&&a.y<b.y); }
const double eps=1e-10;
inline int dcmp(double x)
{ return (x>0?x:-x)<=eps?0:(x>0?1:-1); }
inline bool operator==(const Point &a,const Point &b)
{ return dcmp(a.x-b.x)==0&&dcmp(a.y-b.y)==0; }
inline double Dot(const Vector &A,const Vector &B)
{ return A.x*B.x+A.y*B.y; }
inline double Length(const Vector &A)
{ return sqrt(Dot(A,A)); }
inline double Angle(const Vector &A,const Vector &B)
{ return acos(Dot(A,B)/Length(A)/Length(B)); }
const double Pi=atan(1)*4;
inline double Angle(const Vector &A)
{ return atan2(A.y,A.x); }
inline double R_to_D(double rad)
{ return 180/Pi*rad; }
inline double D_to_R(double D)
{ return Pi/180*D; }
inline double Cross(const Vector &A,const Vector &B)
{ return A.x*B.y-A.y*B.x; }
inline double Area2(const Point &a,const Point &b,const Point &c)
{ return Cross(b-a,c-a); }
inline Vector Rotate(const Vector &A,double rad)
{
double csr=cos(rad),sir=sin(rad);
return Vector(A.x*csr-A.y*sir,A.x*sir+A.y*csr);
}
inline Vector Normal(const Vector &A)
{
double L=Length(A);
return Vector(-A.y/L,A.x/L);
}
inline Vector Format(const Vector &A)
{
double L=Length(A);
return Vector(A.x/L,A.y/L);
}
inline Point GetLineIntersection
(const Point &A1,const Point &B1,const Point &A2,const Point &B2)
{
Vector u=A1-A2,v=B1-A1,w=B2-A2;
double t=Cross(w,u)/Cross(v,w);
return A1+v*t;
}
inline double DistanceToLine
(const Point &P,const Point &A,const Point &B)
{
Vector v1=B-A,v2=P-A;
return fabs(Cross(v1,v2))/Length(v1);
}
inline double DistanceToSegment
(const Point &P,const Point &A,const 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);
}
inline Point GetLineProjection
(const Point &P,const Point &A,const Point &B)
{
Vector v=B-A;
return A+v*(Dot(v,P-A)/Dot(v,v));
}
inline bool SegmentProperIntersection
(const Point &a1,const Point &a2,const Point &b1,const 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;
}
inline bool OnSegment
(const Point &p,const Point &a1,const Point &a2)
{ return dcmp(Cross(a1-p,a2-p))==0&&dcmp(Dot(a1-p,a2-p))<0; }
inline double PolygonArea(Point* p,int n)
{
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;
}
struct Circle
{
Point c;
double r;
Circle(Point c=Point(),double r=0):c(c),r(r){}
inline Point point(double a)
{ return Point(c.x+cos(a)*r,c.y+sin(a)*r); }
};
inline Circle read_circle()
{
Circle C;
scanf("%lf%lf%lf",&C.c.x,&C.c.y,&C.r);
return C;
}
inline int GetLineCircleIntersection
(const Point &A,const Point &B,const Circle &C,vector<Point> &sol)
{
double d=DistanceToLine(C.c,A,B);
int mode=dcmp(d-C.r);
if(mode>0) return 0;//相离
Point P=GetLineProjection(C.c,A,B);
if(mode==0)//相切
{
sol.push_back(P);
return 1;
}
double dist=sqrt(C.r*C.r-d*d);
Vector v=Format(B-A);
sol.push_back(P-v*dist);
sol.push_back(P+v*dist);
return 2;
}
inline int GetCircleCircleIntersection
(Circle C1,Circle C2,vector<Point> &sol)
{
if(C1.r<C2.r) swap(C1,C2);//to make sure C1 is bigger than C2
double D=Length(C1.c-C2.c);
if(dcmp(D)==0)
return dcmp(C1.r-C2.r)==0?-1:0;
if(dcmp(C1.r+C2.r-D)<0) return 0;
if(dcmp(fabs(C1.r-C2.r)-D)>0) return 0;
double d1=((C1.r*C1.r-C2.r*C2.r)/D+D)/2;
double x=sqrt(C1.r*C1.r-d1*d1);
Point O=C1.c+Format(C2.c-C1.c)*d1;
Point P1=O+Normal(O-C2.c)*x,P2=O-Normal(O-C2.c)*x;
sol.push_back(P1);
if(P1==P2) return 1;
sol.push_back(P2);
return 2;
}
inline int GetTangents
(const Point P,const Circle C,vector<Point> &v)
{
Vector u=C.c-P;
double dist=Length(u);
int mode=dcmp(dist-C.r);
if(mode<0) return 0;
if(mode==0)
{
v.push_back(P+Normal(u));
return 1;
}
double x=sqrt(dist*dist-C.r*C.r);
Circle C2(P,x);
return GetCircleCircleIntersection(C,C2,v);
}
using namespace std;
const char* q1="CircumscribedCircle";
const char* q2="InscribedCircle";
const char* q3="TangentLineThroughPoint";
const char* q4="CircleThroughAPointAndTangentToALineWithRadius";
const char* q5="CircleTangentToTwoLinesWithRadius";
const char* q6="CircleTangentToTwoDisjointCirclesWithRadius";
char cmd[100];
Circle WaiJieYuan(Point p1,Point p2,Point p3)
{
double Bx=p2.x-p1.x,By=p2.y-p1.y;
double Cx=p3.x-p1.x,Cy=p3.y-p1.y;
double D=2*(Bx*Cy-By*Cx);
double ansx=(Cy*(Bx*Bx+By*By)-By*(Cx*Cx+Cy*Cy))/D+p1.x;
double ansy=(Bx*(Cx*Cx+Cy*Cy)-Cx*(Bx*Bx+By*By))/D+p1.y;
Point p(ansx,ansy);
return Circle(p,Length(p1-p));
}
Circle NeiJieYuan(Point p1,Point p2,Point p3)
{
double a=Length(p2-p3);
double b=Length(p3-p1);
double c=Length(p1-p2);
Point p=(p1*a+p2*b+p3*c)/(a+b+c);
return Circle(p,DistanceToLine(p,p1,p2));
}
void output(vector<double> res)
{
sort(res.begin(),res.end());
printf("[");
for(int i=0;i<res.size();i++)
{
if(i) printf(",");
printf("%.6lf",res[i]);
}
printf("]
");
}
void output(vector<Point> res)
{
sort(res.begin(),res.end());
printf("[");
for(int i=0;i<res.size();i++)
{
if(i) printf(",");
printf("(%.6lf,%.6lf)",res[i].x,res[i].y);
}
printf("]
");
}
int main()
{
#ifdef local
freopen("pro.in","r",stdin);
#endif
while(scanf("%s",cmd)==1)
{
if(strcmp(cmd,q1)==0)
{
Point p1,p2,p3;
p1=read_point();
p2=read_point();
p3=read_point();
Circle C=WaiJieYuan(p1,p2,p3);
printf("(%.6lf,%.6lf,%.6lf)
",C.c.x,C.c.y,C.r);
}
if(strcmp(cmd,q2)==0)
{
Point p1,p2,p3;
p1=read_point();
p2=read_point();
p3=read_point();
Circle C=NeiJieYuan(p1,p2,p3);
printf("(%.6lf,%.6lf,%.6lf)
",C.c.x,C.c.y,C.r);
}
if(strcmp(cmd,q3)==0)
{
Circle C=read_circle();
Point p=read_point();
vector<Point> v;
vector<double> res;
GetTangents(p,C,v);
for(int i=0;i<v.size();i++)
{
double tmp=R_to_D(Angle(v[i]-p));
if(tmp<0) tmp+=180;
if(tmp>=180) tmp-=180;
res.push_back(tmp);
}
output(res);
}
if(strcmp(cmd,q4)==0)
{
Point p=read_point();
Point A=read_point();
Point B=read_point();
double r;
scanf("%lf",&r);
Circle C(p,r);
Vector v=Normal(B-A)*r;//向两侧平移r
Point A1=A+v,B1=B+v;
Point A2=A-v,B2=B-v;
vector<Point> res;
GetLineCircleIntersection(A1,B1,C,res);
GetLineCircleIntersection(A2,B2,C,res);
output(res);
}
if(strcmp(cmd,q5)==0)
{
Point p1=read_point();
Point p2=read_point();
Point p3=read_point();
Point p4=read_point();
double r;
scanf("%lf",&r);
Vector v=Normal(p1-p2)*r;
Point A1=p1+v,B1=p2+v;
Point A2=p1-v,B2=p2-v;
v=Normal(p3-p4)*r;
Point A3=p3+v,B3=p4+v;
Point A4=p3-v,B4=p4-v;//向两侧平移r
vector<Point> res;
res.push_back(GetLineIntersection(A1,B1,A3,B3));
res.push_back(GetLineIntersection(A1,B1,A4,B4));
res.push_back(GetLineIntersection(A2,B2,A3,B3));
res.push_back(GetLineIntersection(A2,B2,A4,B4));
output(res);
}
if(strcmp(cmd,q6)==0)
{
Circle c1=read_circle();
Circle c2=read_circle();
double r;
scanf("%lf",&r);
c1.r+=r;c2.r+=r;//向外膨胀r
vector<Point> res;
GetCircleCircleIntersection(c1,c2,res);
output(res);
}
}
return 0;
}