POJ 2225 / ZOJ 1438 / UVA 1438 Asteroids --三维凸包，求多面体重心
题意： 两个凸多面体，可以任意摆放，最多贴着，问他们重心的最短距离。
解法： 由于给出的是凸多面体，先构出两个三维凸包，再求其重心，求重心仿照求三角形重心的方式，然后再求两个多面体的重心到每个多面体的各个面的最短距离，然后最短距离相加即为答案，因为显然贴着最优。
求三角形重心见此： http://www.cnblogs.com/whatbeg/p/4234518.html
代码：（模板借鉴网上模板）
#include <iostream> #include <cstdio> #include <cstring> #include <cstdlib> #include <cmath> #include <algorithm> #include <string> #include <vector> #include <set> #define Mod 1000000007 #define eps 1e-8 #define lll __int64 #define ll long long using namespace std; #define N 100007 #define MAXV 505 //三维点 struct pt{ double x, y, z; pt(){} pt(double _x, double _y, double _z): x(_x), y(_y), z(_z){} pt operator - (const pt p1){return pt(x - p1.x, y - p1.y, z - p1.z);} pt operator * (pt p){return pt(y*p.z-z*p.y, z*p.x-x*p.z, x*p.y-y*p.x);} //叉乘 double operator ^ (pt p){return x*p.x+y*p.y+z*p.z;} //点乘 }; //pt operator - (const pt p,const pt p1){return pt(p.x - p1.x, p.y - p1.y, p.z - p1.z);} //pt operator ** (pt p,pt p1){return pt(p.y*p1.z-p.z*p1.y, p.z*p1.x-p.x*p1.z, p.x*p1.y-p.y*p1.x);} //叉乘 //double operator ^^ (pt p1,pt p){return p1.x*p.x+p1.y*p.y+p1.z*p.z;} struct _3DCH{ struct fac{ int a, b, c; //表示凸包一个面上三个点的编号 bool ok; //表示该面是否属于最终凸包中的面 }; int n; //初始点数 pt P[MAXV]; //初始点 int cnt; //凸包表面的三角形数 fac F[MAXV*8]; //凸包表面的三角形 int to[MAXV][MAXV]; double vlen(pt a){return sqrt(a.x*a.x+a.y*a.y+a.z*a.z);} //向量长度 double area(pt a, pt b, pt c){return vlen((b-a)*(c-a));} //三角形面积*2 double volume(pt a, pt b, pt c, pt d){return (b-a)*(c-a)^(d-a);} //四面体有向体积*6 //正：点在面同向 double ptof(pt &p, fac &f){ pt m = P[f.b]-P[f.a], n = P[f.c]-P[f.a], t = p-P[f.a]; return (m * n) ^ t; } pt pvec(fac s) { pt k1 = (P[s.a]-P[s.b]), k2 = (P[s.b]-P[s.c]); return (k1*k2); } double ptoplane(pt p,fac s){ return fabs(pvec(s)^(p-P[s.a]))/vlen(pvec(s)); } void deal(int p, int a, int b){ int f = to[a][b]; fac add; if (F[f].ok){ if (ptof(P[p], F[f]) > eps) dfs(p, f); else{ add.a = b, add.b = a, add.c = p, add.ok = 1; to[p][b] = to[a][p] = to[b][a] = cnt; F[cnt++] = add; } } } void dfs(int p, int cur){ F[cur].ok = 0; deal(p, F[cur].b, F[cur].a); deal(p, F[cur].c, F[cur].b); deal(p, F[cur].a, F[cur].c); } bool same(int s, int t){ pt &a = P[F[s].a], &b = P[F[s].b], &c = P[F[s].c]; return fabs(volume(a, b, c, P[F[t].a])) < eps && fabs(volume(a, b, c, P[F[t].b])) < eps && fabs(volume(a, b, c, P[F[t].c])) < eps; } //构建三维凸包 void construct(){ cnt = 0; if (n < 4) return; /*********此段是为了保证前四个点不公面，若已保证，可去掉********/ bool sb = 1; //使前两点不公点 for (int i = 1; i < n; i++){ if (vlen(P[0] - P[i]) > eps){ swap(P[1], P[i]); sb = 0; break; } } if (sb)return; sb = 1; //使前三点不公线 for (int i = 2; i < n; i++){ if (vlen((P[0] - P[1]) * (P[1] - P[i])) > eps){ swap(P[2], P[i]); sb = 0; break; } } if (sb)return; sb = 1; //使前四点不共面 for (int i = 3; i < n; i++){ if (fabs((P[0] - P[1]) * (P[1] - P[2]) ^ (P[0] - P[i])) > eps){ swap(P[3], P[i]); sb = 0; break; } } if (sb)return; /*********此段是为了保证前四个点不公面********/ fac add; for (int i = 0; i < 4; i++){ add.a = (i+1)%4, add.b = (i+2)%4, add.c = (i+3)%4, add.ok = 1; if (ptof(P[i], add) > 0) swap(add.b, add.c); to[add.a][add.b] = to[add.b][add.c] = to[add.c][add.a] = cnt; F[cnt++] = add; } for (int i = 4; i < n; i++){ for (int j = 0; j < cnt; j++){ if (F[j].ok && ptof(P[i], F[j]) > eps){ dfs(i, j); break; } } } int tmp = cnt; cnt = 0; for (int i = 0; i < tmp; i++){ if (F[i].ok){ F[cnt++] = F[i]; } } } //表面积 double area(){ double ret = 0.0; for (int i = 0; i < cnt; i++){ ret += area(P[F[i].a], P[F[i].b], P[F[i].c]); } return ret / 2.0; } //体积 double volume(){ pt O(0, 0, 0); double ret = 0.0; for (int i = 0; i < cnt; i++) { ret += volume(O, P[F[i].a], P[F[i].b], P[F[i].c]); } return fabs(ret / 6.0); } pt BaryCenter() { pt O(0, 0, 0); double ret = 0.0,sumvolume = 0.0, sumx = 0.0, sumy = 0.0, sumz = 0.0; for(int i=0;i<cnt;i++) { double Vol = volume(O, P[F[i].a], P[F[i].b], P[F[i].c]); sumvolume += Vol; sumx += (P[F[i].a].x + P[F[i].b].x + P[F[i].c].x)*Vol; sumy += (P[F[i].a].y + P[F[i].b].y + P[F[i].c].y)*Vol; sumz += (P[F[i].a].z + P[F[i].b].z + P[F[i].c].z)*Vol; } return pt(sumx/sumvolume/4, sumy/sumvolume/4, sumz/sumvolume/4); } //表面三角形数 int facetCnt_tri(){ return cnt; } //表面多边形数 int facetCnt(){ int ans = 0; for (int i = 0; i < cnt; i++){ bool nb = 1; for (int j = 0; j < i; j++){ if (same(i, j)){ nb = 0; break; } } ans += nb; } return ans; } }; _3DCH hull,hull2; //内有大数组，不易放在函数内 int main() { while (scanf("%d", &hull.n)!=EOF){ for (int i = 0; i < hull.n; i++) scanf("%lf%lf%lf", &hull.P[i].x, &hull.P[i].y, &hull.P[i].z); hull.construct(); pt bc1 = hull.BaryCenter(); scanf("%d",&hull2.n); for (int i = 0; i < hull2.n; i++) scanf("%lf%lf%lf", &hull2.P[i].x, &hull2.P[i].y, &hull2.P[i].z); hull2.construct(); pt bc2 = hull2.BaryCenter(); //printf("BARY1: %.2f %.2f %.2f\n",bc1.x,bc1.y,bc1.z); //printf("BARY2: %.2f %.2f %.2f\n",bc2.x,bc2.y,bc2.z); double dis1 = Mod, dis2 = Mod; for (int i = 0; i < hull.cnt; i++) dis1 = min(dis1,fabs(hull.ptoplane(bc1,hull.F[i]))); for (int i = 0; i < hull2.cnt; i++) dis2 = min(dis2,fabs(hull2.ptoplane(bc2,hull2.F[i]))); printf("%.6f\n",dis1+dis2); } return 0; }