6-5 Construct an animated gif file showing the double generation of the cardioid.

> restart:

> with(plots):

Warning, the name changecoords has been redefined

> x1:=cos(2*t):y1:=sin(2*t):

> x2:=2*cos(t):y2:=2*sin(t):

> x:=x1+x2:y:=y1+y2:

> a:=plot([x,y,t=0..2*Pi],color=blue,scaling=constrained,axes=none):

> a1:=animate([x1+2*cos(s),y1+2*sin(s),s=0..2*Pi],t=0..2*Pi,color=yellow,scaling=constrained,axes=none,frames=50,thickness=2):

> a11:=animate([(1-s)*x1+s*x,(1-s)*y1+s*y,s=0..1],t=0..2*Pi,color=gold,scaling=constrained,axes=none,frames=50,thickness=2):

> a2:=animate([x2+cos(s),y2+sin(s),s=0..2*Pi],t=0..2*Pi,color=red,scaling=constrained,axes=none,frames=50,thickness=2):

> a21:=animate([(1-s)*x2+s*x,(1-s)*y2+s*y,s=0..1],t=0..2*Pi,color=red,scaling=constrained,axes=none,frames=50,thickness=2):

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

> display(a,a1,a11,a2,a21,b);

6 Construct an animated gif file showing the double generation of the nephroid.

> restart:

> with(plots):

Warning, the name changecoords has been redefined

> x1:=cos(3*t):y1:=sin(3*t):

> x2:=3*cos(t):y2:=3*sin(t):

> x:=x1+x2:y:=y1+y2:

> a:=plot([x,y,t=0..2*Pi],color=yellow,scaling=constrained,axes=none):

> a1:=animate([x1+3*cos(s),y1+3*sin(s),s=0..2*Pi],t=0..2*Pi,color=blue,scaling=constrained,axes=none,frames=50,thickness=2):

> a11:=animate([(1-s)*x1+s*x,(1-s)*y1+s*y,s=0..1],t=0..2*Pi,color=blue,scaling=constrained,axes=none,frames=50,thickness=2):

> a2:=animate([x2+cos(s),y2+sin(s),s=0..2*Pi],t=0..2*Pi,color=pink,scaling=constrained,axes=none,frames=50,thickness=2):

> a21:=animate([(1-s)*x2+s*x,(1-s)*y2+s*y,s=0..1],t=0..2*Pi,color=pink,scaling=constrained,axes=none,frames=50,thickness=2):

> b:=plot([2*cos(t),2*sin(t),t=0..2*Pi],color=red,scaling=constrained,axes=none):

> display(a,a1,a11,a2,a21,b);

7 Construct an animated gif file showing the double generation of the deltoid. .

> restart:

> with(plots):

Warning, the name changecoords has been redefined

> x1:=cos(-2*t):y1:=sin(-2*t):

> x2:=2*cos(t):y2:=2*sin(t):

> x:=x1+x2:y:=y1+y2:

> a:=plot([x,y,t=0..2*Pi],color=grey,scaling=constrained,axes=none):

> a1:=animate([x1+2*cos(s),y1+2*sin(s),s=0..2*Pi],t=0..2*Pi,color=blue,scaling=constrained,axes=none,frames=50,thickness=2):

> a11:=animate([(1-s)*x1+s*x,(1-s)*y1+s*y,s=0..1],t=0..2*Pi,color=blue,scaling=constrained,axes=none,frames=50,thickness=2):

> a2:=animate([x2+cos(s),y2+sin(s),s=0..2*Pi],t=0..2*Pi,color=red,scaling=constrained,axes=none,frames=50,thickness=2):

> a21:=animate([(1-s)*x2+s*x,(1-s)*y2+s*y,s=0..1],t=0..2*Pi,color=red,scaling=constrained,axes=none,frames=50,thickness=2):

> b:=plot([3*cos(t),3*sin(t),t=0..2*Pi],color=gold,scaling=constrained,axes=none):

> display(a,a1,a11,a2,a21,b);

8 Construct an animated gif file showing the double generation of the astroid.

> restart:

> with(plots):

Warning, the name changecoords has been redefined

> x1:=cos(-3*t):y1:=sin(-3*t):

> x2:=3*cos(t):y2:=3*sin(t):

> x:=x1+x2:y:=y1+y2:

> a:=plot([x,y,t=0..2*Pi],color=yellow,scaling=constrained,axes=none):

> a1:=animate([x1+3*cos(s),y1+3*sin(s),s=0..2*Pi],t=0..2*Pi,color=blue,scaling=constrained,axes=none,frames=50,thickness=2):

> a11:=animate([(1-s)*x1+s*x,(1-s)*y1+s*y,s=0..1],t=0..2*Pi,color=blue,scaling=constrained,axes=none,frames=50,thickness=2):

> a2:=animate([x2+cos(s),y2+sin(s),s=0..2*Pi],t=0..2*Pi,color=gold,scaling=constrained,axes=none,frames=50,thickness=2):

> a21:=animate([(1-s)*x2+s*x,(1-s)*y2+s*y,s=0..1],t=0..2*Pi,color=gold,scaling=constrained,axes=none,frames=50,thickness=2):

> b:=plot([4*cos(t),4*sin(t),t=0..2*Pi],color=red,scaling=constrained,axes=none):

> display(a,a1,a11,a2,a21,b);

>

>

>