2017 ACM-ICPC 亚洲区（南宁赛区）网络赛 G. Finding the Radius for an Inserted Circle(计算几何，二分)

左手的ㄟ右手 2022-06-08 11:40 198阅读 0赞

**Finding the Radius for an Inserted Circle**

Three circles C\_\{a\}*C**a*, C\_\{b\}*C**b*, and C\_\{c\}*C**c*, all with radius R*R* and tangent to each other, are located in two-dimensional space as shown in Figure 11. A smaller circle C\_\{1\}*C*1 with radius R\_\{1\}*R*1 (R\_\{1\}<R*R*1<*R*) is then inserted into the blank area bounded by C\_\{a\}*C**a*, C\_\{b\}*C**b*, and C\_\{c\}*C**c* so that C\_\{1\}*C*1 is tangent to the three outer circles, C\_\{a\}*C**a*, C\_\{b\}*C**b*, and C\_\{c\}*C**c*. Now, we keep inserting a number of smaller and smaller circles C\_\{k\}\\ (2 \\leq k \\leq N)*C**k* (2≤*k*≤*N*) with the corresponding radius R\_\{k\}*R**k* into the blank area bounded by C\_\{a\}*C**a*, C\_\{c\}*C**c* and C\_\{k-1\}*C**k*−1 (2 \\leq k \\leq N)(2≤*k*≤*N*), so that every time when the insertion occurs, the inserted circle C\_\{k\}*C**k* is always tangent to the three outer circles C\_\{a\}*C**a*, C\_\{c\}*C**c* and C\_\{k-1\}*C**k*−1, as shown in Figure 11

[![1944ff32865c0d3d9750980922f3811d1d3b81f0.png][]][1944ff32865c0d3d9750980922f3811d1d3b81f0.png]

Figure 1.

(Left) Inserting a smaller circle C\_\{1\}*C*1 into a blank area bounded by the circle C\_\{a\}*C**a*, C\_\{b\}*C**b* and C\_\{c\}*C**c*.

(Right) An enlarged view of inserting a smaller and smaller circle C\_\{k\}*C**k* into a blank area bounded by C\_\{a\}*C**a*, C\_\{c\}*C**c* and C\_\{k-1\}*C**k*−1 (2 \\leq k \\leq N2≤*k*≤*N*), so that the inserted circle C\_\{k\}*C**k* is always tangent to the three outer circles, C\_\{a\}*C**a*, C\_\{c\}*C**c*, and C\_\{k-1\}*C**k*−1.

Now, given the parameters R*R* and k*k*, please write a program to calculate the value of R\_\{k\}*R**k*, i.e., the radius of the k-th*k*−*t**h* inserted circle. Please note that since the value of R\_k*R**k* may not be an integer, you only need to report theinteger part of R\_\{k\}*R**k*. For example, if you find that R\_\{k\}*R**k* = 1259.89981259.8998 for some k*k*, then the answer you should report is 12591259.

Another example, if R\_\{k\}*R**k* = 39.102939.1029 for some k*k*, then the answer you should report is 3939.

Assume that the total number of the inserted circles is no more than 1010, i.e., N \\leq 10*N*≤10. Furthermore, you may assume \\pi = 3.14159*π*=3.14159. The range of each parameter is as below:

1 \\leq k \\leq N1≤*k*≤*N*, and 10^\{4\} \\leq R \\leq 10^\{7\}104≤*R*≤107.

### Input Format ###

Contains l + 3*l*\+3 lines.

Line 11: l*l* ----------------- the number of test cases, l*l* is an integer.

Line 22: R*R* ---------------- R*R* is a an integer followed by a decimal point,then followed by a digit.

Line 33: k*k* ---------------- test case \#11, k*k* is an integer.

\\ldots…

Line i+2*i*\+2: k*k* ----------------- test case \# i*i*.

\\ldots…

Line l +2*l*\+2: k*k* ------------ test case \#l*l*.

Line l + 3*l*\+3: \-1−1 ---------- a constant \-1−1 representing the end of the input file.

### Output Format ###

Contains l*l* lines.

Line 11: k*k* R\_\{k\}*R**k* ----------------output for the value of k*k* and R\_\{k\}*R**k* at the test case \#11, each of which should be separated by a blank.

\\ldots…

Line i*i*: k*k* R\_\{k\}*R**k* ----------------output for k*k* and the value of R\_\{k\}*R**k* at the test case \# i*i*, each of which should be separated by a blank.

Line l*l*: k*k* R\_\{k\}*R**k* ----------------output for k*k* and the value ofR\_\{k\}*R**k* at the test case \# l*l*, each of which should be separated by a blank.

#### 样例输入 ####

1
    152973.6
    1
    -1

#### 样例输出 ####

1 23665

题意：这道题就是给你三个一模一样的圆，左边一个右边一个下面一个，三个圆两两相切，然后在这三个圆中间的地方找一个小圆，要求这个小圆与这三个大圆相切，这被称为一次操作，然后第二次操作的时候，就把下面的大圆替换成那个小圆，然后从这两个大圆和一个小圆中间再找一个更小的圆也与它们相切，最后问你经过k次操作后那个小圆的半径是多少。

思路：我们可以把三个圆的圆心两两相连，连成一个三角形，再把三个圆心与中间的小圆心相连，假设左右两个大圆的半径为R，下面的圆半径为r，中间的小圆半径为s，那么就可以转化成下图。

![Center][]

红色的边都是已知长度的，并且长度都在图中标记了出来，因为相切，所以两圆心的距离等于两圆半径之和。

那么我们可以对图中的蓝色边做文章，蓝色边的长度可以等于sqrt((R+s)^2-R^2)，也可以等于sqrt((R+r)^2-R^2)-r-s。这样我们让这两个等式相等，就可以通过R和r求出s了，不过这个方程组里有根号，不是很好解，所以我们可以用二分的方法去寻找方程的解。s求出来后我们就完成了一次操作，然后把s的值赋给r，我们就可以进行第二次操作了。

#include <iostream>  
    #include <cstdio>  
    #include <cstring>  
    #include <string>  
    #include <cstdlib>  
    #include <cmath>  
    #include <vector>  
    #include <queue>  
    #include <map>  
    #include <algorithm>  
    #include <set>  
    #include <functional>  
    using namespace std;
    typedef long long LL;
    typedef unsigned long long ULL;
    const int INF = 1e9 + 5;
    const int MAXN = 100000007;
    const int MOD = 1e9 + 7;
    const double eps = 1e-8;
    const double PI = acos(-1.0);
    LL gcd(LL a, LL b) { return b == 0 ? a : gcd(b, a%b); }
    
    double R,r;
    double get(double x){return sqrt(x*x + 2 * R*x) + x;}
    
    double find(double r)//二分查找
    {
    	int k = 200;
    	double l=0, rr=1e7, m;
    	double aim = sqrt((R + r)*(R + r) - R*R) - r;
    	while (k--)
    	{
    		m = (l + rr) / 2;
    		if (get(m) - aim > eps)
    			rr = m;
    		else
    			l = m;
    	}
    	return m;
    }
    
    int main()
    {
    	int T;
    	int k;
    	scanf("%d", &T);
    	scanf("%lf", &R);
    	while (T--)
    	{
    		scanf("%d", &k);
    		r = R;
    		for (int i = 1; i <= k; i++)
    			r = find(r);
    		printf("%d %d\n", k, (int)r);
    	}
    	scanf("%d", &k);
    }

[1944ff32865c0d3d9750980922f3811d1d3b81f0.png]: /images/20220608/dda99cc35b20415fbaeb3282aa0f4b48.png
[Center]: /images/20220608/203c3d3ab71a4b55a5ebb4b80b1e2ce9.png