restart;with( plots ):with( plottools ):Shrinking Sphere ProblemDerivation of General Formula for Intersection of S and S_r,and Its Projection from the Top of S_r onto the z=0 PlaneWhen S is a ConeDouglas B. Meade9 February 2007
<Text-field style="Heading 1" layout="Heading 1">Initial Configuration: the S (cone) and S_r (sphere) and the point P</Text-field>S := (a,b) -> x^2+(y-a)^2=(z-b)^2; # fixed surfaceSr := r -> x^2+y^2+z^2=r^2; # shrinking sphereP := r -> [ 0, 0, r ]; # top of shrinking sphereplotP := r -> plot3d( P(r), x=-1..1, y=-1..1, style=point, symbol=circle, symbolsize=10, color=blue ):plotS := (a,b) -> implicitplot3d( S(a,b), x= -2*b..2*b, y=a-2*b..a+2*b, z=-b..b,
color=pink, style=patchnogrid, transparency=0.8, grid=[25,25,25] ):plotSr := r -> implicitplot3d( Sr(r), x=-r..r, y=-r..r, z=-r..r,
color=cyan, style=patchnogrid, transparency=0.8 ):P1 := (r,a,b) -> display( [plotP(r),plotS(a,b),plotSr(r)],
axes=normal, labels=["x","y","z"], orientation=[25,65], args[4..-1] ):P1(1,1,1, scaling=constrained);
<Text-field style="Heading 1" layout="Heading 1">Construction of Q: Intersection of S and S_r</Text-field>The intersection between these two spheres is a circle, parallel to the x=0 plane.Intersection := [allvalues( solve( {S(a,b),Sr(r)}, {x,y,z} ) )] ;There are two parts to this solution.xPOS := p->evalb( eval(x, eval(p,[r=1.,a=1.,b=1.,z=0.]))>0 ):Q := [ unapply( eval( [x,y,z], select( xPOS, Intersection )[] ), [z,r,a,b] ), unapply( eval( [x,y,z], remove( xPOS, Intersection )[] ), [z,r,a,b] ) ]:piecewise( x>0, Q[1](z,r,a,b), x<0, Q[2](z,r,a,b) );To construct the projection from the top of the shrinking sphere through Q onto the z=0 plane, the parameterization of Q can be done in terms of z, ranging from the lowest value of 0 to the maximum value ofzM := (r,a,b) -> b*r/sqrt(a^2+b^2);(and back down to zero). Actually, the intersection exists for all z from -zM to zM (with x>0) and back down to -zM (with x<0)This curve is not easily identified.plotQ := (r,a,b) -> display( [seq( spacecurve( Q[i](z,r,a,b), z=-zM(r,a,b)..zM(r,a,b), color=gold, thickness=2 ), i=1..2 )] ):P2 := (r,a,b) -> display( [plotP(r),plotS(a,b),plotSr(r),plotQ(r,a,b)],
axes=normal, labels=["x","y","z"], orientation=[45,60], scaling=constrained ):P2(1,1,1);
<Text-field style="Heading 1" layout="Heading 1">Construction of R: Projection of Q, from P, onto z=0 plane</Text-field>For each angle theta, the lines passing through P and the point Q(theta) can be parameterized in terms of the (scaled) distance measured along this line. LinePQ := [seq( unapply( expand( (1-alpha)*P(r) + alpha*Q[i](z,r,a,b) ), [alpha,z,r,a,b] ), i=1..2 )]:piecewise( x>0, LinePQ[1](alpha,z,r,a,b), x<0, LinePQ[2](alpha,z,r,a,b) );The value of the parameter alpha when these lines hit the z=0 plane are given byalpha0 := unapply( [simplify( solve( LinePQ[1](alpha,z,r,a,b)[3]=0, alpha ) ) assuming a>0, r>0][],
[z,r,a,b] );Thus, the parametric representation of of the projected curve, R, in the z=0 plane isR := [seq( unapply( [simplify( LinePQ[i](alpha0(z,r,a,b),z,r,a,b) ) assuming a>0, b>0, r>0][], [z,r,a,b] ), i=1..2 )]:piecewise( x>0, R[1](z,r,a,b), x<0, R[2](z,r,a,b) );This completes the constructions needed to put all of this together in one animation.plotR := (r,a,b) -> display( [seq( spacecurve( R[i](z,r,a,b), z=-zM(r,a,b).. zM(r,a,b), numpoints=201, color=red, thickness=1 ), i=1..2 )] ):P3 := (r,a,b) -> display( [P2(r,a,b),plotR(r,a,b)] ):animQ := (r,a,b) -> display( [seq( animate( spacecurve, [LinePQ[i](alpha,z,r,a,b), alpha=0..alpha0(z,r,a,b)], z=-zM(r,a,b).. zM(r,a,b),
color=blue, thickness=2, orientation=[25,65], background=P3(r,a,b),
scaling=constrained, frames=21 ), i=1..2 )] ):P3(1/2,1,1);animQ(1,1,1);animQ(1/2,1,1);
<Text-field style="Heading 1" layout="Heading 1">Limit as r -> 0</Text-field>These plots already illustrate the rapid convergence of every point on the curves R - except the one on the x-axis - to the origin (as r->0). Let's look at the parametric form of R. The three components are (for x>0):X,Y,Z := R[2](zeta,r,a,b)[]:
x=X;
y=Y;
z=Z;Notice that, since a<>0, the maximum value of the parameter is strictly less than r. Hence, there are no indeterminate forms, and each of the above expressions converges to 0 (as r->0).map( limit, [X,Y,Z], r=0, right );We close with an animation that shows this convergence.#to3d := transform( (x,y)->[x,y,0] ):#plotR0 := a -> to3d( implicitplot( x^2 + (y-2*a)^2 = 4*a^2, x=-2*a..2*a, y=0..4*a, color=green ) ):animR := (a,b) -> animate( P3, [1-r,a,b], r=0..1, frames=20, numpoints=201, paraminfo=false ):#, background=plotR0(a) ):animR(1,1); # Be patient! This animation could take a while to create.