Wannafly Winter Camp 2019 Day2 H Cosmic Cleaner (球体相交体积(球冠体积公式))

https://zhixincode.com/contest/9/problem/H
题意：在三维空间给定n个球体（圆心坐标和半径），给定以原点为圆心的球的半径，求该球与这n个球体有多少公共体积。
https://blog.csdn.net/enterprise_/article/details/81624174?tdsourcetag=s_pctim_aiomsg
↑↑↑参考该大神的博客可求出相交的两个球体的公共体积，然后特判一下包含和不相交的情况即可。
重点是球冠体积公式：$$V=\pi H^2(R?\frac{H}{3})$$
其中H是球冠的高，R是球冠的半径。

#include <iostream>
#include <cstdio>
#include <algorithm>
#include <cmath>
#include <cstring>
using namespace std;
const double PI = acos(-1.0);
struct point {
    double x,y,z;point() {
    }point(double a, double b,double c) {
    x = a;y = b;z = c;}point operator -(const point &b)const {
         //返回减去后的新点return point(x - b.x, y - b.y,z-b.z);}point operator +(const point &b)const {
         //返回加上后的新点return point(x + b.x, y + b.y,z+b.z);}//数乘计算point operator *(const double &k)const {
        //返回相乘后的新点return point(x * k, y * k,z*k);}point operator /(const double &k)const {
        //返回相除后的新点return point(x / k, y / k,z/k);}double operator *(const point &b)const {
        //点乘return x*b.x + y*b.y+z*b.z;}
};
double dist(point p1, point p2) {
           //返回平面上两点距离return sqrt((p1 - p2)*(p1 - p2));
}
struct sphere {
    //球double r;point centre;sphere() {
    }sphere(point c,double rr) {
    centre=c;r=rr;}
}arr[110];
double SphereInterV(sphere a, sphere b) {
    double d = dist(a.centre, b.centre);//球心距double l1 = ((a.r*a.r - b.r*b.r) / d + d) / 2.0;double l2 = d - l1;double x1 = a.r - l1, x2 = b.r - l2;//分别为两个球缺的高度double v1 = PI*x1*x1*(a.r - x1 / 3.0);//相交部分r1圆所对应的球缺部分体积double v2 = PI*x2*x2*(b.r - x2 / 3.0);//相交部分r2圆所对应的球缺部分体积double v = v1 + v2;//相交部分体积return v;
}int n,T;
int main()
{
    int t=1;scanf("%d",&T);while(T--){
    memset(arr,0,sizeof(arr));double xx,yy,zz,re,ans=0;scanf("%d",&n);for(int i=1;i<=n;i++){
    scanf("%lf%lf%lf%lf",&xx,&yy,&zz,&re);point p(xx,yy,zz);sphere sp(p,re);arr[i]=sp;}scanf("%lf%lf%lf%lf",&xx,&yy,&zz,&re);point pc(xx,yy,zz);sphere sc(pc,re);for(int i=1;i<=n;i++){
    if(dist(sc.centre,arr[i].centre)+arr[i].r<=sc.r) ans+=4.0/3.0*PI*arr[i].r*arr[i].r*arr[i].r;else if(dist(sc.centre,arr[i].centre)>=arr[i].r+sc.r) ans+=0;else ans+=SphereInterV(sc,arr[i]);}printf("Case #%d: %.20lf\n",t,ans);t++;}return 0;
}