HDU4773： Problem of Apollonius

Problem Description

　　Apollonius of Perga (ca. 262 BC - ca. 190 BC) was a Greek geometer and astronomer. In his noted work Epaphai, he posed and solved such a problem: constructing circles that are tangent to three given circles in a plane. Two tangent circles can be internally or externally tangent to each other, thus Apollonius’s problem generically have eight solutions.

　　Now considering a simplified case of Apollonius’s problem: constructing circles that are externally tangent to two given circles, and touches a given point(the given point must be on the circle which you find, can’t be inside the circle). In addition, two given circles have no common points, and neither of them are contained by the other, and the given point is also located strictly outside the given circles. You should be thankful that modern mathematics provides you with plenty of useful tools other than euclidean geometry that help you a lot in this problem.

　　

Input

　　The first line of input contains an integer T (T ≤ 200), indicating the number of cases.

　　Each ease has eight positive integers x1, y1, r1, x2, y2, r2, x3, y3 in a single line, stating two circles whose centres are (x1, y1), (x2, y2) and radius are r1 and r2 respectively, and a point located at (x3, y3). All integers are no larger than one hundred.

Output

　　For each case, firstly output an integer S, indicating the number of solutions.

　　Then output S lines, each line contains three float numbers x, y and r, meaning that a circle, whose center is (x, y) and radius is r, is a solution to this case. If there are multiple solutions (S > 1), outputing them in any order is OK. Your answer will be accepted if your absolute error for each number is no more than 10-4.

Sample Input

1

12 10 1 8 10 1 10 10

Sample Output

2

10.00000000 8.50000000 1.50000000

10.00000000 11.50000000 1.50000000

HINT

我们不难发现本题的做法其实就是你以p为反演中心，随便一个r为反演半径，然后我们再把两个圆反演一波，求一波过p的外公切线，然后再把这个切线反演回来就好。

CODE

#include <bits/stdc++.h>

#define eps 0.00000001
double x_0, y_0;
double a[5], b[5], c[5];
double ansx[5], ansy[5], ansr[5];
int flag[5];

void exchange(double &x, double &y)
{
double tmp = x;
x = y;
y = tmp;
return ;
}

double dis(double x_1, double y_1, double x_2, double y_2)
{
return sqrt((x_1 - x_2) * (x_1 - x_2) + (y_1 - y_2) * (y_1 - y_2));
}

void changepoint(double x, double y, double &ex, double &ey)
{
double d = dis(x, y, x_0, y_0);
ey = y_0 + (y - y_0) / (d * d);
ex = x_0 + (x - x_0) / (d * d);
return ;
}

void changecircle(double x, double y, double r, double &ex, double &ey, double &er)
{
double x_1, y_1, x_2, y_2;
if(fabs(x - x_0) < eps)
{
changepoint(x, y + r, x_1, y_1);
changepoint(x, y- r, x_2, y_2);
}
else
{
double k = (y - y_0) / (x - x_0);
double tmp = sqrt(r * r / (1 + k * k));
changepoint(tmp + x, k * tmp + y, x_1, y_1);
changepoint(-tmp + x, -k * tmp + y, x_2, y_2);
}
ex = (x_1 + x_2) / 2;
ey = (y_1 + y_2) / 2;
er = dis(x_1, y_1, x_2, y_2) / 2;
return ;
}

void rotate(double s, double t, double &aa, double &bb, double &cc, double sinct, double cosct)
{
double tmpa = aa, tmpb = bb, tmpc = cc;
aa = tmpa * cosct - tmpb * sinct;
bb = tmpa * sinct + tmpb * cosct;
cc = tmpc + tmpa * s * cosct + tmpa * t * sinct - tmpa * s
- tmpb * s * sinct + tmpb * t * cosct - tmpb * t;
return ;
}

void getline(double x_1, double y_1, double r_1, double x_2, double y_2, double r_2)
{
double tmpa, tmpb, tmpc;
tmpa = y_2 - y_1;
tmpb = x_1 - x_2;
tmpc = y_1 * (x_1 - x_2) - x_1 * (y_1 - y_2);
double d, detr;
d = dis(x_1, y_1, x_2, y_2);
detr = r_2 - r_1;
double sinct = fabs(detr) / d;
double cosct = sqrt(d * d - detr * detr) / d;
double aa = tmpa, bb = tmpb, cc = tmpc;
rotate(x_1, y_1, aa, bb, cc, sinct, cosct);
a[0] = aa;
b[0] = bb;
c[0] = cc - r_1 * sqrt(a[0] * a[0] + b[0] * b[0]);
aa = tmpa, bb = tmpb, cc = tmpc;
rotate(x_1, y_1, aa, bb, cc, -sinct, cosct);
a[1] = aa;
b[1] = bb;
c[1] = cc + r_1 * sqrt(a[1] * a[1] + b[1] * b[1]);
return ;
}

void getcircle(double aa, double bb, double cc, int k)
{
if(fabs(aa * x_0 + bb * y_0 - cc) < eps)
{
flag[k] = 0;
return ;
}
double s, t, ex, ey;
s = (aa * cc + x_0 * bb * bb - aa * bb * y_0) / (aa * aa + bb * bb);
t = (bb * cc - x_0 * aa * bb + aa * aa * y_0) / (aa * aa + bb * bb);
changepoint(s, t, ex, ey);
ansx[k] = (x_0 + ex) / 2;
ansy[k] = (y_0 + ey) / 2;
ansr[k] = dis(x_0, y_0, ex, ey) / 2;
return ;
}

double x_00,y_00,x_11,y_11,x_22,y_22,r_11,r_22;
bool check(double x, double y, double r)
{
if (fabs(dis(x,y,x_00,y_00) - r) < eps)
if (fabs(dis(x,y,x_11,y_11) - r - r_11) < eps)
if (fabs(dis(x,y,x_22,y_22) - r - r_22) < eps)
return true;
return false;
}

double get_dis(double aa, double bb, double cc,double x, double y)
{
return fabs((aa*x + bb*y - cc)/sqrt(aa*aa + bb*bb));
}

int main()
{

int T = 100;
double x_1, y_1, r_1, x_2, y_2, r_2;
scanf("%d", &T);
while(T--)
{
scanf("%lf%lf%lf", &x_1, &y_1, &r_1);
scanf("%lf%lf%lf", &x_2, &y_2, &r_2);
scanf("%lf%lf", &x_0, &y_0);
x_00 = x_0;
y_00 = y_0;
x_11 = x_1;
y_11 = y_1;
x_22 = x_2;
y_22 = y_2;
r_11 = r_1;
r_22 = r_2;

double ex_1, ex_2, ey_1, ey_2, er_1, er_2;

changecircle(x_1, y_1, r_1, ex_1, ey_1, er_1);
changecircle(x_2, y_2, r_2, ex_2, ey_2, er_2);
if(er_1 > er_2)
{
exchange(ex_1, ex_2);
exchange(ey_1, ey_2);
exchange(er_1, er_2);
}
getline(ex_1, ey_1, er_1, ex_2, ey_2, er_2);
memset(flag, -1, sizeof(flag));
getcircle(a[0], b[0], c[0], 0);
getcircle(a[1], b[1], c[1], 1);
int ans = 0;
for (int i=0; i < 2;i++)
{
if (check(ansx[i],ansy[i],ansr[i])){
ans++;
flag[i] = 1;
}
}

printf("%d\n", ans);

for(int i = 0; i < 2; i++)
{
if(flag[i] == 1){
printf("%.8lf %.8lf %.8lf\n", ansx[i], ansy[i], ansr[i]);
}
}
}
return 0;
}