> | restart; |
> | with( plots ): |
> | with( plottools ): |
> |
Shrinking Sphere Problem
Derivation of General Formula for Intersction of S and S_r,
and Its Projection from the Top of S_r onto the z=0 Plane
Douglas B. Meade
9 February 2007
> |
Initial Configuration: the spheres S and S_r and the point P
> | S := a -> x^2+(y-a)^2+z^2=a^2; # 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 -> implicitplot3d( S(a), x= -a..a, y=0..2*a, z=-a..a,
color=pink, style=patchnogrid, transparency=0.8 ): |
> | plotSr := r -> implicitplot3d( Sr(r), x=-r..r, y=-r..r, z=-r..r,
color=cyan, style=patchnogrid, transparency=0.8 ): |
> | P1 := (r,a) -> display( [plotP(r),plotS(a),plotSr(r)],
axes=normal, labels=["x","y","z"], orientation=[25,65] ): |
> | P1(1,2); |
> |
Construction of Q: Intersection of S and S_r
The intersection between these two spheres is a circle, parallel to the x=0 plane.
> | Intersection := [allvalues( solve( {S(a),Sr(r)}, {x,y,z} ) )] ; |
The two parts to this solution are the top and bottom of the circle.
> |
> | y1,r1 :=eval( [y, sqrt(x^2+z^2)], Intersection[1] )[]: |
> | y2,r2 :=eval( [y, sqrt(x^2+z^2)], Intersection[2] )[]: |
> | y1=y2; |
> | simplify( r1=r2 ) assuming r>0, a>0; |
> |
This shows that Q is the circle with
.
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To construct the projection from the top of the shrinking sphere through Q onto the z=0 plane, parameterize the circle Q according to the angle made the positive x axis
> | Q := unapply( [ r1*sin(theta), y1, r1*cos(theta) ], [theta,r,a] ): |
> |
> | plotQ := (r,a) -> spacecurve( Q(theta,r,a), theta=0..2*Pi,
color=gold, thickness=2 ): |
> | P2 := (r,a) -> display( [plotP(r),plotS(a),plotSr(r),plotQ(r,a)],
axes=normal, labels=["x","y","z"], orientation=[45,60], scaling=constrained ): |
> | P2(1/2,2); |
> |
Construction of R: Projection of Q, from P, onto z=0 plane
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 := unapply( expand( (1-alpha)*P(r) + alpha*Q(theta,r,a) ), [alpha,theta,r,a] ); |
> |
The value of the parameter alpha when these lines hit the z=0 plane are given by
> | alpha0 := unapply( [simplify( solve( LinePQ(alpha,theta,r,a)[3]=0, alpha ) ) assuming a>0, r>0][],
[theta,r,a] ); |
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Thus, the parametric representation of of the projected curve, R, in the z=0 plane is
> | R := unapply( [simplify( LinePQ(alpha0(theta,r,a),theta,r,a) ) assuming a>0, r>0][], [theta,r,a] ); |
> |
This completes the constructions needed to put all of this together in one animation.
> | plotR := (r,a) -> spacecurve( R(theta,r,a), theta=0..2*Pi, numpoints=201,
color=red, thickness=1 ): |
> | P3 := (r,a) -> display( [P2(r,a),plotR(r,a)] ): |
> | animQ := (r,a) -> animate( spacecurve, [LinePQ(alpha,theta,r,a), alpha=0..alpha0(theta,r,a)], theta=0..2*Pi,
color=blue, thickness=2, orientation=[25,65], background=P3(r,a), scaling=constrained, frames=41 ): |
> | animQ(1,2); |
> | animQ(1/2,2); |
> |
Limit as r -> 0
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:
> | X,Y,Z := R(theta,r,a)[]:
x=X; y=Y; z=Z; |
> |
Whenever , these expression are not indeterminate (as r->0) and so
> | map( limit, [X,Y,Z], r=0, right ); |
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But, Maple misses the special case when :
> | eval( [X,Y,Z], theta=0 ); |
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The remaining limit to be evaluated is the same one that was encountered in the Shrinking Circle Problem.
> | map( limit, , r=0, right ) assuming a>0; |
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These pointwise limits are nice, but they do no good in determining the limiting curve of the projected curves, R.
The graphical evidence suggests that the limiting curve could be a circle. If so, then the pointwise limits tell us the only possible circle will be the circle passing through both [0,0,0] and and lying in the z=0 plane. That is,
,
. To confirm this, the first step is to verify for each positive value of r, the projected curve R is a circle:
> | simplify( X^2 + (Y-2*a)^2 ); |
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Now, as r shrinks to zero, the square of the radius clearly increases to
> | limit( , r=0, right ); |
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We close with a different 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 -> animate( P3, [1-r,a], r=0..1, frames=30, numpoints=401, paraminfo=false, background=plotR0(a) ): |
> | animR(1); |
> |
> |