import graph3; size(469pt); viewportmargin=0; currentprojection=perspective( camera=(25.0851928432063,-30.3337528952473,19.3728775115443), up=Z, target=(-0.590622314050054,0.692357205025578,-0.627122488455679), zoom=1, autoadjust=false); triple f(pair t) { real u=t.x; real v=t.y; real r=2-cos(u); real x=3*cos(u)*(1+sin(u))+r*cos(v)*(u < pi ? cos(u) : -1); real y=8*sin(u)+(u < pi ? r*sin(u)*cos(v) : 0); real z=r*sin(v); return (x,y,z); } surface s=surface(f,(0,0),(2pi,2pi),8,8,Spline); draw(s,lightolive+white,"bottle",render(merge=true)); string lo="$\displaystyle u\in[0,\pi]: \cases{x=3\cos u(1+\sin u)+(2-\cos u)\cos u\cos v,\cr y=8\sin u+(2-\cos u)\sin u\cos v,\cr z=(2-\cos u)\sin v.\cr}$"; string hi="$\displaystyle u\in[\pi,2\pi]:\\\cases{x=3\cos u(1+\sin u)-(2-\cos u)\cos v,\cr y=8\sin u,\cr z=(2-\cos u)\sin v.\cr}$"; real h=0.0125; begingroup3("parametrization"); draw(surface(xscale(-0.38)*yscale(-0.18)*lo,s,0,1.7,h,bottom=false), "[0,pi]"); draw(surface(xscale(0.26)*yscale(0.1)*rotate(90)*hi,s,4.9,1.4,h,bottom=false), "[pi,2pi]"); endgroup3(); begingroup3("boundary"); draw(s.uequals(0),blue+dashed); draw(s.uequals(pi),blue+dashed); endgroup3(); add(new void(frame f, transform3 t, picture pic, projection P) { draw(f,invert(box(min(f,P),max(f,P)),P),"frame"); });