> restart:

> with(plots):

Warning, the name changecoords has been redefined

> x:=t*sin(t):y:=t*cos(t):

> a:=plot([x,y,t=0..12*Pi],scaling=constrained,axes=none,color=blue):

> x1:=diff(x,t):y1:=diff(y,t):

> x2:=diff(x1,t):y2:=diff(y1,t):

> f:=(x1^2+y1^2)/(x1*y2-x2*y1):

> xc:=x-f*y1:yc:=y+f*x1:

> b:=plot([xc,yc,t=0..2*Pi],scaling=constrained,axes=none,color=green):

> r:=(x1^2+y1^2)^(3/2)/abs(x1*y2-x2*y1):

> c:=animate([xc+r*cos(s),yc+r*sin(s),s=0..2*Pi],t=0..12*Pi,
scaling=constrained,axes=none,color=red):

> d:=animate([(1-s)*xc+s*x,(1-s)*yc+s*y,s=0..1],t=0..12*Pi,
scaling=constrained,axes=none,color=black):

> display(a,b,c,d);