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-8;
int dcmp(double x)
{
    if(fabs(x) < eps) return 0;
    else return x < 0 ? -1 : 1;
}

struct Point3
{
    double x, y, z;
    Point3(double x=0, double y=0, double z=0):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) == 0 && dcmp(a.y-b.y) == 0 && dcmp(a.z-b.z) == 0;
}

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[3];
    Face(int a, int b, int c)
    {
        v[0] = a;
        v[1] = b;
        v[2] = c;
    }
    Vector3 Normal(const vector<Point3>& P) const
    {
        return Cross(P[v[1]]-P[v[0]], P[v[2]]-P[v[0]]);
    }
    // f是否能看见P[i]
    int CanSee(const vector<Point3>& P, int i) const
    {
        return Dot(P[i]-P[v[0]], Normal(P)) > 0;
    }
};

// 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 = 0; i < n; i++) vis[i].resize(n);

    vector<Face> cur;
    cur.push_back(Face(0, 1, 2)); // 由于已经进行扰动，前三个点不共线
    cur.push_back(Face(2, 1, 0));
    for(int i = 3; i < n; i++)
    {
        vector<Face> next;
        // 计算每条边的“左面”的可见性
        for(int j = 0; j < cur.size(); j++)
        {
            Face& f = cur[j];
            int res = f.CanSee(P, i);
            if(!res) next.push_back(f);
            for(int k = 0; k < 3; k++) vis[f.v[k]][f.v[(k+1)%3]] = res;
        }
        for(int j = 0; j < cur.size(); j++)
            for(int k = 0; k < 3; k++)
            {
                int a = cur[j].v[k], b = cur[j].v[(k+1)%3];
                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) != 1) return false;
        P.resize(n);
        P2.resize(n);
        for(int i = 0; i < n; i++)
        {
            P[i] = read_point3();
            P2[i] = add_noise(P[i]);
        }
        faces = CH3D(P2);
        return true;
    }

    Point3 centroid()
    {
        Point3 C = P[0];
        double totv = 0;
        Point3 tot(0,0,0);
        for(int i = 0; i < faces.size(); i++)
        {
            Point3 p1 = P[faces[i].v[0]], p2 = P[faces[i].v[1]], p3 = P[faces[i].v[2]];
            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 = 0; i < faces.size(); i++)
        {
            Point3 p1 = P[faces[i].v[0]], p2 = P[faces[i].v[1]], p3 = P[faces[i].v[2]];
            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
", d1+d2);
    }
    return 0;
}