2015
05-23

# The Red Gem

In circle land, in the museum of circles, a grand red circular gem is on display.
The curator has decided to spice up the display, and has placed the gem on a purple circular platform, along with mundane orange circular gems.

Starved citizens of circle land (points) have flocked to see the grand exhibit of the exquisite red gem. They cannot step on the purple exhibit floor, but can only stand on the circumference. Unfortunately, the mundane orange gems block the view of the exquisite red gem. Please help the museum folks determine the proportion of the circumference of the purple platform from which all of the red gem is visible, completely unobstructed by the orange gems.

There will be several test cases in the input. Each test case will begin with a line with five integers:
n p x y r
Where n (1≤n≤100) is the number of orange circles, p (10≤p≤1,000) is the radius of the purple platform, (x,y) is the center of the red gem relative to the center of the purple platform (-1,000≤x,y≤1000), and r (0<r≤1000) is the radius of the red gem. The red gem is guaranteed to lie fully on the purple platform. No part of the red gem will extend past the purple platform.
On each of the next n lines will be three integers:
x y r
which represent the (x,y) center (-1,000≤x,y≤1000) relative to the center of the purple platform, and radius r (0<r≤1000) of each orange gem. As with the red gem, each orange gem is guaranteed to lie entirely on the purple platform. The orange gems will not overlap the red gem, and they will not overlap each other. The input will end with a line with 5 0s.

There will be several test cases in the input. Each test case will begin with a line with five integers:
n p x y r
Where n (1≤n≤100) is the number of orange circles, p (10≤p≤1,000) is the radius of the purple platform, (x,y) is the center of the red gem relative to the center of the purple platform (-1,000≤x,y≤1000), and r (0<r≤1000) is the radius of the red gem. The red gem is guaranteed to lie fully on the purple platform. No part of the red gem will extend past the purple platform.
On each of the next n lines will be three integers:
x y r
which represent the (x,y) center (-1,000≤x,y≤1000) relative to the center of the purple platform, and radius r (0<r≤1000) of each orange gem. As with the red gem, each orange gem is guaranteed to lie entirely on the purple platform. The orange gems will not overlap the red gem, and they will not overlap each other. The input will end with a line with 5 0s.

4 10 0 0 1
5 0 2
0 5 2
-5 0 2
0 -5 2
0 0 0 0 0

0.3082

#include<cstdio>
#include<cstring>
#include<algorithm>
#include<cmath>
#include<vector>
#define eps 1e-8
#define N 200
using namespace std;

const double PI=acos(-1.0);
int dlcmp(double x)
{
return x<-eps?-1:x>eps;
}

struct Point
{
double x,y;

Point(){}
Point(double a,double b):x(a),y(b){}

Point operator + (const Point a) const {return Point(x+a.x,y+a.y);}
Point operator - (const Point a) const {return Point(x-a.x,y-a.y);}
Point operator * (const double a) const {return Point(x*a,y*a);}
Point operator / (const double a) const {return Point(x/a,y/a);}

bool operator == (const Point a)
{
return !dlcmp(x-a.x)&&!dlcmp(y-a.y);
}

Point trunc(double d) const
{
double dis(Point,Point);
double len=dis(*this,Point(0,0));
return Point(x*d/len,y*d/len);
}

Point rotate(double a)
{
return Point(x*cos(a)-y*sin(a),y*cos(a)+x*sin(a));
}
};

struct Circle
{
Point o;
double r;

Circle(){}
Circle(Point a,double l):o(a),r(l){}
};

struct node
{
double s,e;

node(){}
node(double a,double b):s(a),e(b){}
};

Circle c[N],red,purple;
vector<node>arc;

double sqr(double x)
{
return x*x;
}

double fix(double x)
{
if (dlcmp(x-1)>=0)
return 1;
if (dlcmp(x+1)<=0)
return -1;
return x;
}

double dis(Point a,Point b)
{
return sqrt(sqr(a.x-b.x)+sqr(a.y-b.y));
}

void get_InCommonTangent(Circle c1,Circle c2,Point &s1,Point &e1,Point &s2,Point &e2)
{
double l=dis(c1.o,c2.o);
double d=l*c1.r/(c1.r+c2.r);
double tmp=c1.r/d;
tmp=fix(tmp);
double theta=acos(tmp);
Point vec=c2.o-c1.o;

vec=vec.trunc(c1.r);
s1=c1.o+vec.rotate(theta);
s2=c1.o+vec.rotate(-theta);

vec=c1.o-c2.o;
vec=vec.trunc(c2.r);
e1=c2.o+vec.rotate(theta);
e2=c2.o+vec.rotate(-theta);
}

void inter_circle_ray(Circle c,Point s,Point e,Point &p)
{
Point vec=e-s;
double A=sqr(vec.x)+sqr(vec.y);
double B=2*(vec.x*(s.x-c.o.x)+vec.y*(s.y-c.o.y));
double C=sqr(s.x-c.o.x)+sqr(s.y-c.o.y)-sqr(c.r);
double delta=sqr(B)-4*A*C;

delta=fabs(delta);

double k=(-B+sqrt(delta))/(2*A);

p=s+vec*k;
}

double get_angle(Point a)
{
return atan2(a.y,a.x);
}

double cmp(node a,node b)
{
if (dlcmp(a.s-b.s)==0)
return dlcmp(a.e-b.e)<0;
else
return dlcmp(a.s-b.s)<0;
}

int main()
{
int n,p,x,y,r,i,j;
double ans,sum,cur;
node u,v;
Point s1,e1,s2,e2,vec,p1,p2;

while (scanf("%d%d%d%d%d",&n,&p,&x,&y,&r),n||p||x||y||r)
{
purple.o=Point(0,0);
purple.r=(double)p;
red.o.x=(double)x;
red.o.y=(double)y;
red.r=(double)r;

arc.clear();

for (i=0;i<n;i++)
scanf("%lf%lf%lf",&c[i].o.x,&c[i].o.y,&c[i].r);

for (i=0;i<n;i++)
{
get_InCommonTangent(red,c[i],s1,e1,s2,e2);

if (s1==e1)
{
vec=c[i].o-red.o;
e1=s1+vec.rotate(PI/2);
e2=s2+vec.rotate(-PI/2);
}

inter_circle_ray(purple,s1,e1,p1);
inter_circle_ray(purple,s2,e2,p2);

u.s=get_angle(p1);
u.e=get_angle(p2);

if (u.s>u.e)
{
arc.push_back(node(u.s,PI));
arc.push_back(node(-PI,u.e));
}
else
arc.push_back(u);
}

sort(arc.begin(),arc.end(),cmp);

sum=arc[0].e-arc[0].s;
cur=arc[0].e;

for (i=1;i<arc.size();i++)
{
if (dlcmp(arc[i].e-cur)<=0)
continue;

if (dlcmp(arc[i].s-cur)>0)
sum+=arc[i].e-arc[i].s;
else
sum+=arc[i].e-cur;

cur=arc[i].e;
}

ans=(2*PI-sum)/(2*PI);

if (ans<0)
ans=0;

printf("%.4f\n",ans);
}

return 0;
}