uvalive 4589 Asteroids

题意：给两个凸包，凸包能旋转，求凸包重心之间的最短距离。

思路：显然两个凸包贴在一起时，距离最短。所以，先求重心，再求重心到各个面的最短距离。

三维凸包+重心求法

重心求法：在凸包内，任意枚举一点，在与凸包其他一个面组成一个三棱锥。求出每个三棱锥的重心，把三棱锥等效成一个个质点，再求整体的重心。

 #include<cstdio>
 #include<cmath>
 #include<cstdlib>
 #include<cstring>
 #include<queue>
 using namespace std;

 const double eps = 1e-;
 int dcmp(double x)
 {
     if(fabs(x) < eps) return ;
     else return x <  ? - : ;
 }

 struct Point3
 {
     double x, y, z;
     Point3(double x=, double y=, double z=):x(x),y(y),z(z) { }
 };

 typedef Point3 Vector3;

 Vector3 operator + (const Vector3& A, const Vector3& B)
 {
     return Vector3(A.x+B.x, A.y+B.y, A.z+B.z);
 }
 Vector3 operator - (const Point3& A, const Point3& B)
 {
     return Vector3(A.x-B.x, A.y-B.y, A.z-B.z);
 }
 Vector3 operator * (const Vector3& A, double p)
 {
     return Vector3(A.x*p, A.y*p, A.z*p);
 }
 Vector3 operator / (const Vector3& A, double p)
 {
     return Vector3(A.x/p, A.y/p, A.z/p);
 }

 bool operator == (const Point3& a, const Point3& b)
 {
     return dcmp(a.x-b.x) ==  && dcmp(a.y-b.y) ==  && dcmp(a.z-b.z) == ;
 }

 Point3 read_point3()
 {
     Point3 p;
     scanf("%lf%lf%lf", &p.x, &p.y, &p.z);
     return p;
 }

 double Dot(const Vector3& A, const Vector3& B)
 {
     return A.x*B.x + A.y*B.y + A.z*B.z;
 }
 double Length(const Vector3& A)
 {
     return sqrt(Dot(A, A));
 }
 double Angle(const Vector3& A, const Vector3& B)
 {
     return acos(Dot(A, B) / Length(A) / Length(B));
 }
 Vector3 Cross(const Vector3& A, const Vector3& B)
 {
     return Vector3(A.y*B.z - A.z*B.y, A.z*B.x - A.x*B.z, A.x*B.y - A.y*B.x);
 }
 double Area2(const Point3& A, const Point3& B, const Point3& C)
 {
     return Length(Cross(B-A, C-A));
 }
 double Volume6(const Point3& A, const Point3& B, const Point3& C, const Point3& D)
 {
     return Dot(D-A, Cross(B-A, C-A));
 }
 Point3 Centroid(const Point3& A, const Point3& B, const Point3& C, const Point3& D)
 {
     return (A + B + C + D)/4.0;
 }

 double rand01()
 {
     return rand() / (double)RAND_MAX;
 }
 double randeps()
 {
     return (rand01() - 0.5) * eps;
 }

 Point3 add_noise(const Point3& p)
 {
     return Point3(p.x + randeps(), p.y + randeps(), p.z + randeps());
 }

 struct Face
 {
     int v[];
     Face(int a, int b, int c)
     {
         v[] = a;
         v[] = b;
         v[] = c;
     }
     Vector3 Normal(const vector<Point3>& P) const
     {
         return Cross(P[v[]]-P[v[]], P[v[]]-P[v[]]);
     }
     // f是否能看见P[i]
     int CanSee(const vector<Point3>& P, int i) const
     {
         return Dot(P[i]-P[v[]], Normal(P)) > ;
     }
 };

 // 增量法求三维凸包
 // 注意：没有考虑各种特殊情况（如四点共面）。实践中，请在调用前对输入点进行微小扰动
 vector<Face> CH3D(const vector<Point3>& P)
 {
     int n = P.size();
     vector<vector<int> > vis(n);
     for(int i = ; i < n; i++) vis[i].resize(n);

     vector<Face> cur;
     cur.push_back(Face(, , )); // 由于已经进行扰动，前三个点不共线
     cur.push_back(Face(, , ));
     for(int i = ; i < n; i++)
     {
         vector<Face> next;
         // 计算每条边的“左面”的可见性
         for(int j = ; j < cur.size(); j++)
         {
             Face& f = cur[j];
             int res = f.CanSee(P, i);
             if(!res) next.push_back(f);
             for(int k = ; k < ; k++) vis[f.v[k]][f.v[(k+)%]] = res;
         }
         for(int j = ; j < cur.size(); j++)
             for(int k = ; k < ; k++)
             {
                 int a = cur[j].v[k], b = cur[j].v[(k+)%];
                 if(vis[a][b] != vis[b][a] && vis[a][b]) // (a,b)是分界线，左边对P[i]可见
                     next.push_back(Face(a, b, i));
             }
         cur = next;
     }
     return cur;
 }

 struct ConvexPolyhedron
 {
     int n;
     vector<Point3> P, P2;
     vector<Face> faces;

     bool read()
     {
         if(scanf("%d", &n) != ) return false;
         P.resize(n);
         P2.resize(n);
         for(int i = ; i < n; i++)
         {
             P[i] = read_point3();
             P2[i] = add_noise(P[i]);
         }
         faces = CH3D(P2);
         return true;
     }

     Point3 centroid()
     {
         Point3 C = P[];
         double totv = ;
         Point3 tot(,,);
         for(int i = ; i < faces.size(); i++)
         {
             Point3 p1 = P[faces[i].v[]], p2 = P[faces[i].v[]], p3 = P[faces[i].v[]];
             double v = -Volume6(p1, p2, p3, C);
             totv += v;
             tot = tot + Centroid(p1, p2, p3, C)*v;
         }
         return tot / totv;
     }

     double mindist(Point3 C)
     {
         double ans = 1e30;
         for(int i = ; i < faces.size(); i++)
         {
             Point3 p1 = P[faces[i].v[]], p2 = P[faces[i].v[]], p3 = P[faces[i].v[]];
             ans = min(ans, fabs(-Volume6(p1, p2, p3, C) / Area2(p1, p2, p3)));
         }
         return ans;
     }
 };

 int main()
 {
     int n, m;
     ConvexPolyhedron P1, P2;
     while(P1.read() && P2.read())
     {
         Point3 C1 = P1.centroid();
         double d1 = P1.mindist(C1);
         Point3 C2 = P2.centroid();
         double d2 = P2.mindist(C2);
         printf("%.8lf\n", d1+d2);
     }
     return ;
 }