sphere-sphere.tex

\documentclass{article}
\usepackage{lmodern}
\usepackage[inline]{asymptote}
\begin{document}
\begin{figure}
\begin{asy}
settings.prc=false;
settings.render=0;
import graph3;

currentprojection=orthographic(camera=(150,900,412),up=(-0.4,-0.7,1.4),target=(100,111,1),zoom=0.5);
currentlight=nolight;
size(300);size3(300);

real r1=162;
real r2=100;

triple v1=(0,0,0);
triple v2=(250,0,0);

// from http://mathworld.wolfram.com/Sphere-SphereIntersection.html :
real d=arclength(v1--v2); // distance between v1,v2
real a=1/2/d*sqrt((2*d*r1)^2-(d^2-r2^2+r1^2)^2);

triple P=(sqrt(r1^2-a^2),0,-a);

triple fs1(pair t){
  return v1+r1*(cos(t.x)*sin(t.y),sin(t.x)*sin(t.y),cos(t.y));
}

triple fs2(pair t){
  return v2+r2*(cos(t.x)*sin(t.y),sin(t.x)*sin(t.y),cos(t.y));
}

surface s1=surface(fs1,(0,0),(2pi,pi),16,Spline);
surface s2=surface(fs2,(0,0),(2pi,pi),16,Spline);

draw(s1
  ,darkgreen+opacity(0.2)
  ,render(compression=Low,merge=true)
);

draw(s2
  ,darkblue+opacity(0.2)
  ,render(compression=Low,merge=true)
);

dot(v1); label("$V_1$",v1,-X+Z);
dot(v2); label("$V_2$",v2,-X+Z);
dot(P); label("$P$",P,-X-Z);
\end{asy}
\end{figure}
\end{document}