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 ;
}