restart; with( plots ): with( plottools ): Shrinking Sphere Problem Derivation of General Formula for Intersection of S and S_r, and Its Projection from the Top of S_r onto the z=0 Plane When S is an Ellipsoid Douglas B. Meade 9 February 2007

S := (a,b,c) -> (x/a)^2+((y-b)/b)^2+(z/c)^2=1; # fixed surface Sr := r -> x^2+y^2+z^2=r^2; # shrinking sphere P := r -> [ 0, 0, r ]; # top of shrinking sphere plotP := r -> plot3d( P(r), x=-1..1, y=-1..1, style=point, symbol=circle, symbolsize=10, color=blue ): plotS := (a,b,c) -> implicitplot3d( S(a,b,c), x= -a..a, y=0..2*b, z=-c..c, 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,c) -> display( [plotP(r),plotS(a,b,c),plotSr(r)], axes=normal, labels=["x","y","z"], orientation=[25,65], args[5..-1] ): P1(1,1,2,3, scaling=constrained);

The intersection between these two spheres is a circle, parallel to the x=0 plane. Intersection := allvalues( solve( {S(a,b,c),Sr(r)}, {x,y} ) ): simplify( [Intersection] ) assuming a>b, b>c, c>0; There are two parts to this solution. xPOS := (p,q)->evalb( eval(x, eval(p,[r=1.,a=1.,b=1/2.,c=2.,z=1/2.]))>0 ) = q: Q := [ unapply( eval( [x,y,z], select( xPOS, [Intersection], true )[] ), [z,r,a,b,c] ), unapply( eval( [x,y,z], select( xPOS, [Intersection], false )[] ), [z,r,a,b,c] ) ]: piecewise( x>0, Q[1](y,r,a,b,c), x<0, Q[2](y,r,a,b,c) ); 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. The minimum and maximum values of z occur when x=0. We now find the highest point, which we call Qstar. yzPOS := (p,q) -> evalb( eval([y,z], eval(p,[r=1.,a=1.,b=1.,c=2.]))::[positive,positive] ) = q: Q0 := allvalues( solve( eval({S(a,b,c),Sr(r)},x=0), {y,z} ) ): Qstar := unapply( eval( [0,y,z], select( yzPOS, [Q0], true )[] ), [r,a,b,c] ): yM := unapply( Qstar(r,a,b,c)[2], [r,a,b,c] ); zM := unapply( Qstar(r,a,b,c)[3], [r,a,b,c] ); This curve is not easily identified. plotQ := (r,a,b,c) -> display( [seq( spacecurve( Q[i](z,r,a,b,c), z=-zM(r,a,b,c)..zM(r,a,b,c), color=gold, thickness=2 ), i=1..2 )] ): P2 := (r,a,b,c) -> display( [plotP(r),plotS(a,b,c),plotSr(r),plotQ(r,a,b,c)], axes=normal, labels=["x","y","z"], orientation=[45,60], scaling=constrained ): P2(1,3,2,1);

For each value of z, the lines passing through P and the corresponding point on Q 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,c) ), [alpha,z,r,a,b,c] ), i=1..2 )]: piecewise( x>0, LinePQ[1](alpha,z,r,a,b,c), x<0, LinePQ[2](alpha,z,r,a,b,c) ); The value of the parameter alpha when these lines hit the z=0 plane are given by alpha0 := unapply( [simplify( solve( LinePQ[1](alpha,z,r,a,b,c)[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 is R := [seq( unapply( LinePQ[i](alpha0(z,r,a,b,c),z,r,a,b,c), [z,r,a,b,c] ), i=1..2 )]: map( simplify, piecewise( x>0, R[1](z,r,a,b,c), x<0, R[2](z,r,a,b,c) ) ); This completes the constructions needed to put all of this together in one animation. plotR := (r,a,b,c) -> display( [seq( spacecurve( R[i](z,r,a,b,c), z=-zM(r,a,b,c).. zM(r,a,b,c), numpoints=201, color=red, thickness=1 ), i=1..2 )] ): P3 := (r,a,b,c) -> display( [P2(r,a,b,c),plotR(r,a,b,c)] ): animQ := (r,a,b,c) -> display( [seq( animate( spacecurve, [LinePQ[i](alpha,z,r,a,b,c), alpha=0..alpha0(z,r,a,b,c)], z=-zM(r,a,b,c).. zM(r,a,b,c), color=blue, thickness=2, orientation=[25,65], background=P3(r,a,b,c), scaling=constrained, frames=41 ), i=1..2 )] ): P3(1,3,2,1); animQ( 1 ,3,2,1); animQ(1/2,3,2,1); animQ(1/4,3,2,1);

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[1](zeta,r,a,b,c)[]: x=X; y=Y; z=Z; It is not surprising that Maple reports that every point converges to the origin. map( limit, [X,Y,Z], r=0, right ); We know there might be something interesting happening at the highest points on Q (along the positive y-axis). These points are given by Qstar := eval( [X,Y,Z], zeta=zM(r,a,b,c) ): These expressions do not simplify much, but we do see that the z-component is 0. simplify( Qstar ) assuming a>b,b>c,c>0; In the limit, we expect that x approaches zero and y approaches twice the radius of the limiting circle. The general limit is still the origin map( limit, Qstar, r=0, right ); but, at the highest point on Q: map( limit, Qstar, r=0, right ) assuming a>b,b>c,c>0; This suggests that the limiting curve approached by R is the circle centered at [ 0, 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0 ] with radius 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. Notice that the curvature of the restriction of S to the y=0 plane is 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 .This result is consistent with the general conclusion (with NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZHLyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYlRiwtRiQ2Ji1GLTY5USRyaG9GKEYwRjNGNi9GOkY4RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbi9Gam5RJ25vcm1hbEYoRlxvLUkjbW9HRiU2M1EiPUYoLyUlZm9ybUdRJmluZml4RigvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRigvJSdyc3BhY2VHRmVwLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGWC8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRy1JJm1mcmFjR0YlNiotRiQ2Iy1JI21uR0YlNjlGYXFGMEYzRjZGZW9GPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmZvRlxvLUYkNiMtRi02OVEma2FwcGFGKEYwRjNGNkZlb0Y8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GZm9GXG8vJS5saW5ldGhpY2tuZXNzR1EiMUYoLyUrZGVub21hbGlnbkdRJ2NlbnRlckYoLyUpbnVtYWxpZ25HRmJzLyUpYmV2ZWxsZWRHRjhGXHJGXnJGLEYsRiw3Iy9JJHJob0dGKCokSSZrYXBwYUdGKCEiIg== = 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). We close with an animation that shows this convergence. to3d := transform( (x,y)->[x,y,0] ): plotR0 := p -> to3d( implicitplot( x^2 + (y-2*p)^2 = (2*p)^2, x=-2*p..2*p, y=0..4*p, color=green ) ): animR := (a,b,c) -> animate( P3, [1-r,a,b,c], r=0..1, frames=40, numpoints=401, paraminfo=false, background=plotR0(c^2/b) ): animR(2,1.5,1); # Be patient! This animation could take a while to create.