• db cir()
  计算多边形周长
• db dir_area()
  计算多边形有向面积
• db area()
  计算多边形面积
• void fix()
  如果点是以顺时针顺序存储，则改为逆时针
• int poly::point_in(pt t)
  判断点和多边形的关系：0 在外面，1 在内部，2 在边上
• int poly::Border_pt_num()
  计算多边形边上的整点个数
• int poly::Inside_pt_num()
  计算多边形内部点的个数
• pt poly::MassCenter()
  计算多边形的重心

namespace Poly {
    struct poly {
        vector<pt> a;
        poly(int sz = 0) {
            a.resize(sz);
        }
        db cir() {
            db res = 0; int n = (int)a.size();
            for (int i = 0; i < n; i++) res += dist(a[i], a[nex(i)]);
            return res;
        }
        db dir_area() {
            db sum = 0; int n = (int)a.size();
            for (int i = 0; i < n; i++) sum += det(a[i], a[nex(i)]);
            return (sum / 2.0);
        }
        db area() {
            return fabs(dir_area());
        }
        void fix() {
            if (cmp(dir_area()) < 0) reverse(a.begin(), a.end());
        }
        int point_in(pt t);
        int Border_pt_num();
        int Inside_pt_num();
        pt MassCenter();
    };

    int poly::point_in(pt t) {
        int num = 0, n = (int)a.size(), d1, d2, k;
        for (int i = 0; i < n; i++) {
            if (Line::pt_on_seg(t, a[i], a[nex(i)])) return 2;
            k = cmp(det(a[nex(i)] - a[i], t - a[i]));
            d1 = cmp(a[i].y - t.y);
            d2 = cmp(a[nex(i)].y - t.y);
            if (k > 0 && d1 <= 0 && d2 > 0) num++;
            if (k < 0 && d2 <= 0 && d1 > 0) num--;
        }
        return num != 0;
    }
    int poly::Border_pt_num() {
        int num = 0, n = (int)a.size();
        for (int i = 0; i < n; i++) {
            int xx = int(fabs(a[nex(i)].x - a[i].x) + 0.01);
            int yy = int(fabs(a[nex(i)].y - a[i].y) + 0.01);
            num += gcd(xx, yy);
        }
        return num;
    }
    int poly::Inside_pt_num() {
        return int(area() + 1 - Border_pt_num() / 2);
    }
    pt poly::MassCenter() {
        int n = (int)a.size();
        pt ans = pt(0, 0);
        if(cmp(area()) == 0) return ans;
        for(int i = 0; i < n; i++)
            ans = ans + (a[i] + a[nex(i)]) * det(a[i], a[nex(i)]);
        return ans / dir_area() / 6.0;
    }
}
using namespace Poly;