[HDU 3623 / TJU 3740][坐标旋转](2010天津网络赛C题)

/*
题目：Covering Points
题目来源：HDU 3623 / TJU 3740 (2010天津网络赛C题)
题目内容或思路：
　　坐标旋转
　　其实这题感觉上也不是太大自然，不过不知道为什么比赛的时候过队的比较少。
　　最让我兴奋的是此题居然是1A的。
　　不过遗憾的是先看了Roba的解题报告。
　　思路在Roba大牛的blog上有http://roba.rushcj.com/?p=523#comments
做题日期：2011.3.4
*/
#include <cstdio>
#include <cstdlib>
#include <climits>
#include <iostream>
#include <algorithm>
#include <cstring>
#include <string>
#include <queue>
#include <map>
#include <vector>
#include <bitset>
#include <cmath>
#include <set>
#include <utility>
using namespace std;

const double eps = 1e-9;
const double inf = 1e100;
const double pi = acos(-1.0);
const int N = 10010;
int n, m;
double ang[N];
double h0, h1, w0, w1, xl, xr, yl, yr;

struct cpoint {
     double x, y;
     cpoint() {}
     cpoint(double x, double y) : x(x), y(y) {}
}cp[N], ch[N], bp;

int dcmp(double x) {
     if (x < -eps) return -1; else return x > eps;
}

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

double dissqr(cpoint p1, cpoint p2) {
     return sqrt(sqr(p1.x - p2.x) + sqr(p1.y - p2.y));
}

double cross(cpoint p0, cpoint p1, cpoint p2) {
     return (p1.x - p0.x) * (p2.y - p0.y) - (p2.x - p0.x) * (p1.y - p0.y);
}

bool PointEqual(cpoint a, cpoint b) {
     return dcmp(a.x - b.x) == 0 && dcmp(a.y - b.y) == 0;
}

int PolarCmp(const cpoint &p1, const cpoint &p2) {
     int u = dcmp(cross(bp, p1, p2));
     return u > 0 || (u == 0 && dcmp(dissqr(bp, p1) - dissqr(bp, p2)) < 0 );
}

void graham(cpoint pin[], int n, cpoint ch[], int &m) {
     int i, j, k, u, v;
     memcpy(ch, pin, n * sizeof(cpoint));
     for (i = k = 0; i < n; ++i) {
         u = dcmp(ch[i].x - ch[k].x);
         v = dcmp(ch[i].y - ch[k].y);
         if (v < 0 || (v == 0 && u < 0)) k = i;
     }
     bp = ch[k];
     sort(ch, ch + n, PolarCmp);
     n = unique(ch, ch + n, PointEqual) - ch;
     if (n <= 1) {m = n; return;}
     if (dcmp(cross(ch[0], ch[1], ch[n - 1])) == 0) {
         m = 2; ch[1] = ch[n - 1]; return;
     }
     ch[n++] = ch[0];
     for (i = 1, j = 2; j < n; ++j) {
         while (i > 0 && dcmp(cross(ch[i - 1], ch[i], ch[j])) <= 0)i--;
         ch[++i] = ch[j];
     }
     m = i;
}

cpoint rotate(cpoint v, double angle) {
     cpoint res;
     double c, s;
     c = cos(angle);
     s = sin(angle);
     res.x = v.x * c - v.y * s;
     res.y = v.x * s + v.y * c;
     return res;
}

void rotate(cpoint ch[], cpoint cp[], int m, double angle) {
     xl = yl = inf;
     xr = yr = -inf;
     for (int i = 0; i < m; ++i) {
         cp[i] = rotate(ch[i], angle);
         xl = min(xl, cp[i].x);
         xr = max(xr, cp[i].x);
         yl = min(yl, cp[i].y);
         yr = max(yr, cp[i].y);
     }
     w1 = xr - xl;
     h1 = yr - yl;
}

int check(double angle) {
     rotate(ch, cp, m, angle);
     if (dcmp(w1 - h1) == 0) return 0;
     if (dcmp(w0 - h0) * dcmp(w1 - h1) <= 0) return -1;
     return 1;
}

void solve() {
     double x, y;
     int i;
     for (i = 0; i < n; ++i) {
         scanf("%lf%lf", &cp[i].x, &cp[i].y);
     }
     graham(cp, n, ch, m);
     if (m < 3) {
         printf("%.8lf %.8lf\n", ch[0].x, ch[0].y);
         x = ( ch[0].x + ch[1].x - ch[0].y + ch[1].y) / 2;
         y = ( ch[0].x - ch[1].x + ch[0].y + ch[1].y) / 2;
         printf("%.8lf %.8lf\n", x, y);
         printf("%.8lf %.8lf\n", ch[1].x, ch[1].y);
         x = ( ch[0].x + ch[1].x + ch[0].y - ch[1].y) / 2;
         y = (-ch[0].x + ch[1].x + ch[0].y + ch[1].y) / 2;
         printf("%.8lf %.8lf\n", x, y);
         return;
     }
     n = 0;
     ch[m] = ch[0];
     for (i = 0; i < m; ++i) {
         ang[n++] = -atan2(ch[i + 1].y - ch[i].y, ch[i + 1].x - ch[i].x);
     }
     sort(ang, ang + n);
     n = unique(ang, ang + n) - ang;
     for (i = 0; i < n; ++i) {
         rotate(ch, cp, m, ang[i]);
         if (i && dcmp(w0 - h0) * dcmp(w1 - h1) <= 0) break;
         w0 = w1;
         h0 = h1;
     }
     double low, high, mid;
     if (i == n) {
         low = ang[n - 1];
         high = ang[0] + pi * 2;
     } else {
         low = ang[i - 1];
         high = ang[i];
     }
     while (1) {
         mid = (low + high) / 2;
         int t = check(mid);
         if (t == 0) break;
         if (t == 1) low = mid;
         else high = mid;
     }
     bp = rotate(cpoint(xl, yl), -mid);
     printf("%.8lf %.8lf\n", bp.x, bp.y);
     bp = rotate(cpoint(xr, yl), -mid);
     printf("%.8lf %.8lf\n", bp.x, bp.y);
     bp = rotate(cpoint(xr, yr), -mid);
     printf("%.8lf %.8lf\n", bp.x, bp.y);
     bp = rotate(cpoint(xl, yr), -mid);
     printf("%.8lf %.8lf\n", bp.x, bp.y);
}

int main() {
#ifndef ONLINE_JUDGE
    freopen("D:\\in.txt", "r", stdin);
#endif
     int cas = 0;
     while (scanf("%d", &n), n) {
         if (cas) puts("");
         printf("Case %d:\n", ++cas);
         solve();
     }
    return 0;
}