> restart;m:=[[-2,sin(t)],[cos(t),sin(t)],[cos(t)-cos(2*t),sin(t)-sin(2*t)]];
   m := [[-2, sin(t)], [cos(t), sin(t)],
> t:=n*2*Pi/100;
> plot([m$n=-25..25],color=cyan,scaling=constrained,axes=none);

> m:=[[1,0],[cos(t),sin(t)],[cos(2*t),sin(2*t)]];
> t:=n*2*Pi/100;
> plot([m$n=0..100],color=blue,scaling=constrained,axes=none);

> restart;
> with(linalg):
> a:=5;b:=3;r:=1.34;s:=2.96;
> ll:=[a*cos(r),b*sin(r)];l:=[a*cos(s),b*sin(s)];
> m:=[l,ll];
> for k to 200 do

    u:=ll-l: n:=[-b*ll[1]/a,-a*ll[2]/b]:v:=u-2*dotprod(u ,   n)*n/dotprod(n,n):

    t:=2*(b^2*v[1]*ll[1]+a^2*ll[2]*v[2])/(b^2*v[1]^2+a^2*v[2]^2):

    lll:=ll+t*v:m:=[op(m),lll]:l:=ll:ll:=lll:od:
> plot(m,scaling=constrained,ax es=none);

> restart;
> with(linalg):
> a:=5;b:=3;r:=1.34;s:=5.88;
> ll:=[a*cos(r),b*sin(r)];l:=[a*cos(s),b*sin(s)];
> m:=[l,ll];
> for k to 200 do

    u:=ll-l: n:=[-b*ll[1]/a,-a*ll[2]/b]:v:=u-2*dotprod(u , n)*n/dotprod(n,n):

    t:=-2*(b^2*v[1]*ll[1]+a^2*ll[2]*v[2])/(b^2*v[1]^2 +a^2*v[2]^2):

    lll:=ll+t*v:m:=[op(m),lll]:l:=ll:ll:=lll:od:
> plot(m,scaling=constrained,axes=none);

> restart;
> k:=3;
> x:=k*cos(t)-cos(k*t);y:=k*sin(t)-sin(k*t);
> 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+0.00000001);
> xc:=x-f*y1;yc:=y+f*x1;
> m:=[[x,y],[xc,yc]];
> t:=n*2*Pi/100;
> plot([m$n=1..100],color=orange,scaling=constrained,axes=none);

> restart;
> k:=-3;
> x:=k*cos(t)-cos(k*t);y:=k*sin(t)-sin(k*t);
> 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+0.00000001);
> xc:=x-f*y1;yc:=y+f*x1;
> m:=[[x,y],[xc,yc]];
> t:=n*2*Pi/100;
> plot([m$n=1..100],color=red,scaling=constrained,axes=none);

> restart;
> k:=-2;
> x:=k*cos(t)-cos(k*t);y:=k*sin(t)-sin(k*t);
> 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+0.00000001);
> xc:=x-f*y1;yc:=y+f*x1;
> m:=[[x,y],[xc,yc]];
> t:=n*2*Pi/100;
> plot([m$n=1..100],color=red,scaling=constrained,axes=none);

> restart;
> k:=2;
> x:=k*cos(t)-cos(k*t);y:=k*sin(t)-sin(k*t);
> 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+0.00000001);
> xc:=x-f*y1;yc:=y+f*x1;
> m:=[[x,y],[xc,yc]];
> t:=n*2*Pi/100;
> plot([m$n=1..100],color=red,scaling=constrained,axes=none);

> restart;
> k:=2;
> x:=k*cos(t)-cos(k*t);y:=k*sin(t)-sin(k*t);
> x1:=diff(x,t);y1:=diff(y,t);
> x2:=diff(x1,t);y2:=diff(y1,t);
> xc:=x-f*y1;yc:=y+f*x1;
> p1:=plot([x,y,t=0..2*Pi],color=yellow,scaling=constrained,axes=none):
> p2:=plot([xc,yc,t=0..2*Pi],

    scaling=constrained,color=green,axes=none):
> r:=(x1^2+y1^2)^(3/2)/abs(x1*y2-y1*x2+0.00000000000001);
> with(plots):
> p3:=animate([xc+r*cos(s),yc+r*sin(s),s=0..2*Pi],t=0..2*Pi,

    color=blue,scaling=constrained):
> p4:=animate([(1-s)*x+s*xc,(1-s)*y+s*yc,s=0..1],t=0..2*Pi,

    color=blue,scaling=constrained):
> display(p1,p2,p3,p4);

> restart;
> k:=3;
> x:=k*cos(t)-cos(k*t);y:=k*sin(t)-sin(k*t);
> 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+0.00000001);
> xc:=x-f*y1;yc:=y+f*x1;
> p1:=plot([x,y,t=0..2*Pi],color=yellow,

   scaling=constrained,axes=none):
> p2:=plot([xc,yc,t=0..2*Pi],

    scaling=constrained,color=green,axes=none):
> r:=(x1^2+y1^2)^(3/2)/abs(x1*y2-y1*x2+0.00000000000001);
> with(plots):
> p3:=animate([xc+r*cos(s),yc+r*sin(s),s=0..2*Pi],t=0..2*Pi,color=blue,

     scaling=constrained):
> p4:=animate([(1-s)*x+s*xc,(1-s)*y+s*yc,s=0..1],t=0..2*Pi,color=blue,

     scaling=constrained):
> display(p1,p2,p3,p4);

> restart;
> k:=-2;
> x:=k*cos(t)-cos(k*t);y:=k*sin(t)-sin(k*t);
> 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+0.00000001);
> xc:=x-f*y1;yc:=y+f*x1;
> p1:=plot([x,y,t=0..2*Pi],color=yellow,

     scaling=constrained,axes=none):
> p2:=plot([xc,yc,t=0..2*Pi],

    scaling=constrained,color=green,axes=none):
> r:=(x1^2+y1^2)^(3/2)/abs(x1*y2-y1*x2+0.00000000000001);
> with(plots):
> p3:=animate([xc+r*cos(s),yc+r*sin(s),s=0..2*Pi],t=0..2*Pi,

    color=blue,scaling=constrained):
> p4:=animate([(1-s)*x+s*xc,(1-s)*y+s*yc,s=0..1],t=0..2*Pi,

    color=blue,scaling=constrained):
> display(p1,p2,p3,p4);

> restart;
> k:=-3;
> x:=k*cos(t)-cos(k*t);y:=k*sin(t)-sin(k*t);
> 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+0.00000001);
> xc:=x-f*y1;yc:=y+f*x1;
> p1:=plot([x,y,t=0..2*Pi],color=yellow,

    scaling=constrained,axes=none):
> p2:=plot([xc,yc,t=0..2*Pi],

    scaling=constrained,color=green,axes=none):
> r:=(x1^2+y1^2)^(3/2)/abs(x1*y2-y1*x2+0.00000000000001);
> with(plots):
> p3:=animate([xc+r*cos(s),yc+r*sin(s),s=0..2*Pi],t=0..2*Pi,

    color=blue,scaling=constrained):
> p4:=animate([(1-s)*x+s*xc,(1-s)*y+s*yc,s=0..1],t=0..2*Pi,

    color=blue,scaling=constrained):
> display(p1,p2,p3,p4);