OrigSC.mw

 > 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

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)=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; >