> | restart; |
> | with( plots ): |
> | with( plottools ): |
Warning, the name changecoords has been redefined
Warning, the assigned name arrow now has a global binding
> |
Shrinking Circle Problem
Solution to the Original Shrinking Circle Problem
Douglas B. Meade
9 February 2007
> |
> | C := (x-1)^2+y^2=1^2; # fixed circle |
> | Cr := r -> x^2+y^2=r^2; # shrinking circle |
> | P := r -> [ 0, r ]; # top of shrinking circle |
> |
> | plotP := r -> plot( [P(r)], x=-1..1, style=point, symbol=circle, symbolsize=10, thickness=3, color=blue ): |
> | plotC := implicitplot( C, x= 0..2, y=-1..1, color=pink ): |
> | plotCr := r -> implicitplot( Cr(r), x=-r..r, y=-r..r, color=cyan ): |
> | P1 := r -> display( [plotP(r),plotC,plotCr(r)], view=[-r..max(r,2),DEFAULT], scaling=constrained,
axes=normal, labels=["x","y"] ): |
> | P1(1,2); |
> |
These circles intersect in two points.
> | Intersection := [allvalues( solve( {C,Cr(r)}, {x,y} ) )] ; |
Of the two points, we want the one in the first quadrant
> | X,Y := eval( [x,y], select( p->evalb( eval( y, eval(p,[r=1.]) )>0 ), Intersection )[] )[]:
x=X; y=Y; |
> |
This is the formula for the point Q. .
> |
To construct the projection from the top of the shrinking circle through Q onto the y=0 plane,
> | Q := unapply( [ X, Y ], [r,a] ): |
> |
> | plotQ := (r) -> plot( [Q(r)], color=gold, style=point, thickness=2 ): |
> | P2 := (r) -> display( [plotP(r),plotC,plotCr(r),plotQ(r)],
axes=normal, labels=["x","y"], scaling=constrained ): |
> | P2(1,2); |
> |
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(r) ), [alpha,r] ); |
> |
The value of the parameter alpha when these lines hit the z=0 plane are given by
> | alpha0 := unapply( [simplify( solve( LinePQ(alpha,r)[2]=0, alpha ) ) assuming r>0][],
[r] ); |
> |
Thus, the parametric representation of of the projected point, R, in the y=0 plane is
> | R := unapply( [simplify( LinePQ(alpha0(r),r) ) assuming r>0][], [r] ); |
> |
This completes the constructions needed to put all of this together in one animation.
> | plotR := (r) -> plot( [P(r),R(r)], color=red, thickness=1 ): |
> | P3 := (r,a) -> display( [P2(r),plotR(r)] ): |
> |
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 := R(r)[]:
x=X; y=Y; |
> |
Maple does recognize the indeterminate form, and reports:
> | limit( X, r=0, right ); |
We close with a different animation that shows this convergence.
> | plotR0 := plot( [[4,0]], x=0..4, color=green, style=point, thickness=2 ): |
> | animR := animate( P3, [2-r], r=0..2, frames=30, numpoints=401, paraminfo=false, background=plotR0, view=[-2..max(2,4),DEFAULT] ): |
> | animR; |
> |