#include<cstdio>
#include<algorithm>
#include<cstring>
#include<queue>
#include<map>
#include<cctype>
#include<cmath>
#include<bitset>
#include<ctime>
#define pprint(x) ::print(x),putchar(' ')
#define fprint(x) ::print(x),putchar('\n')
using namespace std;
//#define getchar() (SS == TT && (TT = (SS = BB) + fread(BB,1,1 << 15,stdin),TT == SS) ? EOF : *SS++)
char BB[1 << 15],*SS = BB,*TT = BB;
using namespace std;
inline int read()
{
int x = 0,f = 1;
char ch = getchar();
for(;!isdigit(ch);ch = getchar())
if(ch == '-')
f = -1;
for(;isdigit(ch);ch = getchar())
x = x * 10 + (ch ^ 48);
return x * f;
}
void print(long long x)
{
if(x < 0)
putchar('-'),x = -x;
if(x > 9)
print(x / 10);
putchar(x % 10 + '0');
}
const double pi = acos(-1),eps = 1e-8;
const int N = 110;
int dcmp(double x)
{
return x < -eps ? -1 : (x > eps);
}
struct point
{
double x,y,z;

void read()
{
scanf("%lf %lf %lf",&x,&y,&z);
}
point(double _x = 0,double _y = 0,double _z = 0)
{
x = _x,y = _y,z = _z;
}
friend point operator + (point szq,point yqy)
{
return point(szq.x + yqy.x,szq.y + yqy.y,szq.z + yqy.z);
}
friend point operator - (point szq,point yqy)
{
return point(szq.x - yqy.x,szq.y - yqy.y,szq.z - yqy.z);
}
friend point operator * (point szq,double yqy)
{
return point(szq.x * yqy,szq.y * yqy,szq.z * yqy);
}
friend double operator * (point szq,point yqy)
{
return szq.x * yqy.x + szq.y * yqy.y + szq.z * yqy.z;
}
friend point operator / (point szq,double yqy)
{
return point(szq.x / yqy,szq.y / yqy,szq.z / yqy);
}
friend point operator & (point szq,point yqy)
{
return point(szq.y * yqy.z - szq.z * yqy.y,szq.z * yqy.x - szq.x * yqy.z,szq.x * yqy.y - szq.y * yqy.x);
}
double lenth()
{
return sqrt((*this) * (*this));
}
};
double angle(point a1,point a2,point b1,point b2)
{
point p = a2 & a1,q = b2 & b1;
return acos((p * q) / p.lenth() / q.lenth());
}
int n,m;
point p[N],G;
int d[N],f[N][N];
int main()
{
n = read(),m = read();
for(int i = 1;i <= n;i++)
p[i].read();
for(int i = 1;i <= m;i++)
{
d[i] = read();
for(int j = 1;j <= d[i];j++)
f[i][j] = read();
}
point u = p[1];
double sum = 0;
for(int i = 1;i <= m;i++)
{
point a = p[f[i][1]],b,c;
for(int j = 2;j < d[i];j++)
{
b = p[f[i][j]],c = p[f[i][j + 1]];
double w = fabs(((b - a) & (c - a)) * (u - a));
G = G + (u + a + b + c) * w * 0.25,sum += w;
}
}
G = G / sum;
for(int i = 1;i <= n;i++)
p[i] = p[i] - G;
double all = 4 * pi;
for(int i = 1;i <= m;i++)
{
double sum = 0;
f[i][0] = f[i][d[i]],f[i][d[i] + 1] = f[i][1];
for(int j = 1;j <= d[i];j++)
sum += angle(p[f[i][j]],p[f[i][j - 1]],p[f[i][j]],p[f[i][j + 1]]);
sum -= (d[i] - 2) * pi;
printf("%.7lf\n",sum / all);
}
return 0;
}

重心的求法没什么好说的。

对于一个多边形，区域 T 是多边形以重心为点光源，在单位球上的投影，该投影是一个曲面多边形。

考虑该曲面多边形的求法，设边数为 d，我们将这个曲面多边形“三角剖分”，那么它的面积就等于所有曲面三角形的面积之和。题目中给出了 \operatorname{Area}(ABC)=\alpha +\beta+\gamma-\piArea(ABC)=α+β+γ−π，所以总面积 S=S= 所有三角形内角和 -(d-2)\pi−(d−2)π，d-2d−2 为三角形个数。

曲面三角形的内角和之和等于曲面多边形内角和，对于曲面多边形一个内角，它等于重心与原多边形一对邻边形成的两个平面的夹角。

最后概率等于区域 T 的面积除以单位球的表面积。

#include<cstdio>
#include<algorithm>
#include<cstring>
#include<queue>
#include<map>
#include<cctype>
#include<cmath>
#include<bitset>
#include<ctime>
#define pprint(x) ::print(x),putchar(' ')
#define fprint(x) ::print(x),putchar('\n')
using namespace std;
//#define getchar() (SS == TT && (TT = (SS = BB) + fread(BB,1,1 << 15,stdin),TT == SS) ? EOF : *SS++)
char BB[1 << 15],*SS = BB,*TT = BB;
using namespace std;
inline int read()
{
int x = 0,f = 1;
char ch = getchar();
for(;!isdigit(ch);ch = getchar())
if(ch == '-')
f = -1;
for(;isdigit(ch);ch = getchar())
x = x * 10 + (ch ^ 48);
return x * f;
}
void print(long long x)
{
if(x < 0)
putchar('-'),x = -x;
if(x > 9)
print(x / 10);
putchar(x % 10 + '0');
}
const double pi = acos(-1),eps = 1e-8;
const int N = 110;
int dcmp(double x)
{
return x < -eps ? -1 : (x > eps);
}
struct point
{
double x,y,z;

void read()
{
scanf("%lf %lf %lf",&x,&y,&z);
}
point(double _x = 0,double _y = 0,double _z = 0)
{
x = _x,y = _y,z = _z;
}
friend point operator + (point szq,point yqy)
{
return point(szq.x + yqy.x,szq.y + yqy.y,szq.z + yqy.z);
}
friend point operator - (point szq,point yqy)
{
return point(szq.x - yqy.x,szq.y - yqy.y,szq.z - yqy.z);
}
friend point operator * (point szq,double yqy)
{
return point(szq.x * yqy,szq.y * yqy,szq.z * yqy);
}
friend double operator * (point szq,point yqy)
{
return szq.x * yqy.x + szq.y * yqy.y + szq.z * yqy.z;
}
friend point operator / (point szq,double yqy)
{
return point(szq.x / yqy,szq.y / yqy,szq.z / yqy);
}
friend point operator & (point szq,point yqy)
{
return point(szq.y * yqy.z - szq.z * yqy.y,szq.z * yqy.x - szq.x * yqy.z,szq.x * yqy.y - szq.y * yqy.x);
}
double lenth()
{
return sqrt((*this) * (*this));
}
};
double angle(point a1,point a2,point b1,point b2)
{
point p = a2 & a1,q = b2 & b1;
return acos((p * q) / p.lenth() / q.lenth());
}
int n,m;
point p[N],G;
int d[N],f[N][N];
int main()
{
n = read(),m = read();
for(int i = 1;i <= n;i++)
p[i].read();
for(int i = 1;i <= m;i++)
{
d[i] = read();
for(int j = 1;j <= d[i];j++)
f[i][j] = read();
}
point u = p[1];
double sum = 0;
for(int i = 1;i <= m;i++)
{
point a = p[f[i][1]],b,c;
for(int j = 2;j < d[i];j++)
{
b = p[f[i][j]],c = p[f[i][j + 1]];
double w = fabs(((b - a) & (c - a)) * (u - a));
G = G + (u + a + b + c) * w * 0.25,sum += w;
}
}
G = G / sum;
for(int i = 1;i <= n;i++)
p[i] = p[i] - G;
double all = 4 * pi;
for(int i = 1;i <= m;i++)
{
double sum = 0;
f[i][0] = f[i][d[i]],f[i][d[i] + 1] = f[i][1];
for(int j = 1;j <= d[i];j++)
sum += angle(p[f[i][j]],p[f[i][j - 1]],p[f[i][j]],p[f[i][j + 1]]);
sum -= (d[i] - 2) * pi;
printf("%.7lf\n",sum / all);
}
return 0;
}