E.球的体积并
题意
求两个球的体积并
做法
用球缺公式+计算几何模板即可解决。
球缺公式为:
其中
是球的半径,
是球缺的高。
代码
#include<stdio.h>
#include<iostream>
#include<algorithm>
#include<math.h>
using namespace std;
const int maxn = 1e5+5;
typedef double db;
const db PI = acos(-1);
const db eps = 1e-10;
int sgn(double x)
{
if(fabs(x)<eps) return 0;
if(x<0) return -1;
else return 1;
}
typedef 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 { return point(x - b.x, y - b.y,z-b.z); }
point operator +(const point &b)const { return point(x + b.x, y + b.y,z+b.z); }
//数乘计算
point operator *(const double &k)const { return point(x * k, y * k,z*k); }
point operator /(const double &k)const { return point(x / k, y / k,z/k); }
double operator *(const point &b)const { return x*b.x + y*b.y+z*b.z; }
}point;
double dist(point p1, point p2) { return sqrt((p1 - p2)*(p1 - p2)); }
struct sphere
{
double r;
point centre;
}sc[maxn];
void SphereInterVS(sphere a, sphere b,double &v)
{
double d = dist(a.centre, b.centre);
if(sgn(d-a.r-b.r)>=0)//相离
{
v=0.0;
return ;
}
else if(sgn(d-fabs(a.r-b.r))<=0)//内含
{
if(sgn(a.r-b.r)<=0) v=4.0*PI*a.r*a.r*a.r/3.0;
else v=4.0*PI*b.r*b.r*b.r/3.0;
return ;
}
//球心距
double t = (d*d + a.r*a.r - b.r*b.r) / (2.0 * d);
//h1=h2,球冠的高
double h = sqrt((a.r*a.r) - (t*t)) * 2;
double angle_a = 2 * acos((a.r*a.r + d*d - b.r*b.r) / (2.0 * a.r*d)); //余弦公式计算r1对应圆心角,弧度
double angle_b = 2 * acos((b.r*b.r + d*d - a.r*a.r) / (2.0 * b.r*d)); //余弦公式计算r2对应圆心角,弧度
double l1 = ((a.r*a.r - b.r*b.r) / d + d) / 2;
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);//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);//相交部分r2圆所对应的球缺部分体积
v = v1 + v2;//相交部分体积
return ;
}
int main()
{
double ans=0;
double x,y,z;
scanf("%lf%lf%lf%lf",&x,&y,&z,&sc[1].r);
point p =point(x,y,z);
sc[1].centre=p;
scanf("%lf%lf%lf%lf",&x,&y,&z,&sc[2].r);
point p2 =point(x,y,z);
sc[2].centre=p2;
ans=ans+4.0/3*PI*sc[1].r*sc[1].r*sc[1].r;
ans=ans+4.0/3*PI*sc[2].r*sc[2].r*sc[2].r;
double v=0.0;
SphereInterVS(sc[1],sc[2],v);
ans=ans-v;
printf("%.10f\n",ans);
return 0;
}