In case you don’t know the definition of convex hull, here we give you a clarification from Wikipedia:
*Convex hull: In mathematics, the convex hull, for a set of points X in a real vector space V, is the minimal convex set containing X.
InputThere are several test cases. In each case the first line contains an integer N indicates the number of 3D-points (3< N <= 300), and then N lines follow, each line contains three numbers x, y, z (between -10000 and 10000) indicate the 3d-position of a point.OutputOutput the number of faces of the 3D-Convex hull.
Sample Input
7 1 1 0 1 -1 0 -1 1 0 -1 -1 0 0 0 1 0 0 0 0 0 -0.1 7 1 1 0 1 -1 0 -1 1 0 -1 -1 0 0 0 1 0 0 0 0 0 0.1
Sample Output
8 5
#include<iostream>
#include<cmath>
#include<cstring>
#include<cstdlib>
#include<algorithm>
#include <stdio.h>
using namespace std;
const int MAXN=1001;
const int N = 500;
const double EPS=1e-8;
int g[MAXN][MAXN];
struct Point{
double x,y,z;
Point(){}
Point(double xx,double yy,double zz):x(xx),y(yy),z(zz){}
Point operator -(const Point p1){//两向量之差
return Point(x-p1.x,y-p1.y,z-p1.z);
}
Point operator *(Point p){//叉乘
return Point(y*p.z-z*p.y,z*p.x-x*p.z,x*p.y-y*p.x);
}
double operator ^(Point p){//点乘
return (x*p.x+y*p.y+z*p.z);
}
};
struct CH3D{
struct face{
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;//初始顶点数
Point P[MAXN];//初始顶点
int num; //凸包表面的三角形数
face F[8*MAXN];
//int g[MAXN][MAXN];//凸包表面的三角形
double vlen(Point a){//向量长度
return sqrt(a.x*a.x+a.y*a.y+a.z*a.z);
}
Point cross(const Point &a, const Point &b, const Point &c){//叉乘
return Point((b.y-a.y)*(c.z-a.z)-(b.z-a.z)*(c.y-a.y),-((b.x-a.x)*(c.z-a.z)-(b.z-a.z)*(c.x-a.x)),(b.x-a.x)*(c.y-a.y)-(b.y-a.y)*(c.x-a.x));
}
double area(Point a,Point b,Point c){//三角形面积*2
return vlen((b-a)*(c-a));
}
double volume(Point a,Point b,Point c,Point d){//四面体有向体积*6
return (b-a)*(c-a)^(d-a);
}
double dblcmp(Point &p,face &f){//正:点在面同向
Point m=P[f.b]-P[f.a];
Point n=P[f.c]-P[f.a];
Point t=p-P[f.a];
return (m*n)^t;
}
void deal(int p,int a,int b){
int f=g[a][b];
face add;
if(F[f].ok){
if(dblcmp(P[p],F[f])>EPS){
dfs(p,f);
}
else {
add.a=b;
add.b=a;
add.c=p;
add.ok=1;
g[p][b]=g[a][p]=g[b][a]=num;
F[num++]=add;
}
}
}
void dfs(int p,int now){
F[now].ok=0;
deal(p,F[now].b,F[now].a);
deal(p,F[now].c,F[now].b);
deal(p,F[now].a,F[now].c);
}
bool same(int s,int t){
Point &a=P[F[s].a];
Point &b=P[F[s].b];
Point &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;
}
void pretreat(){//构建三维凸包
int i,j,tmp;
face add;
bool flag;
num=0;
if(n<4) return;
flag=true;
for(i=1;i<n;i++){//此段是为了保证前四个点不共面,若以保证,则可去掉
if(vlen(P[0]-P[i])>EPS){
swap(P[1],P[i]);
flag=false; break;
}
}
if(flag) return;
flag=true;
for(i=2;i<n;i++){//使前三点不共线
if(vlen((P[0]-P[1])*(P[1]-P[i]))>EPS){
swap(P[2],P[i]);
flag=false; break;
}
}
if(flag) return;
flag=true;
for(i=3;i<n;i++){//使前四点不共面
if(fabs((P[0]-P[1])*(P[1]-P[2])^(P[0]-P[i]))>EPS){
swap(P[3],P[i]);
flag=false;
break;
}
}
if(flag) return;
for(i=0;i<4;i++){
add.a=(i+1)%4;
add.b=(i+2)%4;
add.c=(i+3)%4;
add.ok=true;
if(dblcmp(P[i],add)>0)
swap(add.b,add.c);
g[add.a][add.b]=g[add.b][add.c]=g[add.c][add.a]=num;
F[num++]=add;
}
for(i=4;i<n;i++){
for(j=0;j<num;j++){
if(F[j].ok && dblcmp(P[i],F[j])>EPS){
dfs(i,j);
break;
}
}
}
tmp=num;
for(i=num=0;i<tmp;i++){
if(F[i].ok) F[num++]=F[i];
}
}
int polygon(){//表面多边形个数
int i,j,res,flag;
for(i=res=0;i<num;i++){
flag=1;
for(j=0;j<i;j++)
if(same(i,j)){
flag=0; break;
}
res+=flag;
}
return res;
}
};
int main()
{
int i,n;
while(scanf("%d",&n)!=EOF){
CH3D hull;
memset(g,0,sizeof(g));
memset(hull.P,0,sizeof(hull.P));
hull.n = n;
for(i=0;i<hull.n;i++){
scanf("%lf%lf%lf",&hull.P[i].x,&hull.P[i].y,&hull.P[i].z);
}
hull.pretreat();
printf("%d\n",hull.polygon());
}
return 0;
}