Not So Simple Polygon Embedding CodeForces

Not So Simple Polygon Embedding CodeForces - 1354C2（计算几何）

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The statement of this problem is the same as the statement of problem C1. The only difference is that, in problem C1, n is always even, and in C2, n is always odd.

You are given a regular polygon with 2⋅n vertices (it’s convex and has equal sides and equal angles) and all its sides have length 1. Let’s name it as 2n-gon.

Your task is to find the square of the minimum size such that you can embed 2n-gon in the square. Embedding 2n-gon in the square means that you need to place 2n-gon in the square in such way that each point which lies inside or on a border of 2n-gon should also lie inside or on a border of the square.

You can rotate 2n-gon and/or the square.

Input
The first line contains a single integer T (1≤T≤200) — the number of test cases.

Next T lines contain descriptions of test cases — one per line. Each line contains single odd integer n (3≤n≤199). Don’t forget you need to embed 2n-gon, not an n-gon.

Output
Print T real numbers — one per test case. For each test case, print the minimum length of a side of the square 2n-gon can be embedded in. Your answer will be considered correct if its absolute or relative error doesn’t exceed 10−6.

Example
Input
3
3
5
199
Output
1.931851653
3.196226611
126.687663595
思路：挺难想的一个题目，贴一个参考博客吧：参考博客
代码如下：

#include<bits/stdc++.h>
#define ll long long
#define PI acos(-1.0)
using namespace std;int n;int main()
{int t;scanf("%d",&t);while(t--){scanf("%d",&n);double z=1.0/(2.0-2.0*cos(PI/(n*1.0)));z=sqrt(z);double x1=sin(PI/(n*1.0)*round(n/4.0))*z*sqrt(2.0);double x2=sin(PI/2.0-(PI/(n*1.0)*round(n/4.0)))*z*sqrt(2.0);x1+=x2;printf("%.9lf\n",x1);}return 0;
}