Construct an animated gif file showing the formation of the cardioid.

> restart;

> m1:=plot([cos(t),sin(t),t=0..2*Pi],axes=none,scaling=constrained,color=black):

> m2:=plot([2*cos(u)-cos(2*u),2*sin(u)-sin(2*u),u=0..2*Pi],axes=none,scaling=constrained,color=green):

> with(plots):

Warning, the name changecoords has been redefined

> m3:=animate([v*2*cos(s*Pi/10)+(1-v)*(2*cos(s*Pi/10)-cos(2*s*Pi/10)),v*2*sin(s*Pi/10)+(1-v)*(2*sin(s*Pi/10)-sin(2*s*Pi/10)),v=0..1],s=1..20,axes=none,scaling=constrained,color=red,frames=50):

> m4:=animate([2*cos(s*Pi/10)+cos(t),2*sin(s*Pi/10)+sin(t),t=0..2*Pi],s=1..20,axes=none,scaling=constrained,color=blue,frames=50):

> display(m1,m2,m3,m4);

Construct an animated gif file showing the formation of the nephroid.

> restart;

> m1:=plot([cos(t),sin(t),t=0..2*Pi],axes=none,scaling=constrained,color=black):

> m2:=plot([3/2*cos(u)-cos(3*u)/2,3/2*sin(u)-sin(3*u)/2,u=0..2*Pi],axes=none,scaling=constrained,color=green):

> with(plots):

Warning, the name changecoords has been redefined

> m3:=animate([v*3*cos(u)/2+(1-v)*(3*cos(u)/2-cos(3*u)/2),v*3*sin(u)/2+(1-v)*(3*sin(u)/2-sin(3*u)/2),v=0..1],u=0..2*Pi,axes=none,scaling=constrained,color=red,frames=50):

> m4:=animate([3*cos(u)/2+cos(t)/2,3*sin(u)/2+sin(t)/2,t=0..2*Pi],u=0..2*Pi,axes=none,scaling=constrained,color=blue,frames=50):

> display(m1,m2,m3,m4);

Construct an animated gif file showing the formation of the deltoid .

> restart;

> m1:=plot([cos(t),sin(t),t=0..2*Pi],axes=none,scaling=constrained,color=black):

> m2:=plot([2/3*cos(u)+cos(-2*u)/3,2/3*sin(u)+sin(-2*u)/3,u=0..2*Pi],axes=none,scaling=constrained,color=green):

> with(plots):

Warning, the name changecoords has been redefined

> m3:=animate([v*2*cos(u)/3+(1-v)*(2*cos(u)/3+cos(-2*u)/3),v*2*sin(u)/3+(1-v)*(2*sin(u)/3+sin(-2*u)/3),v=0..1],u=0..2*Pi,axes=none,scaling=constrained,color=red,frames=50):

> m4:=animate([2*cos(u)/3+cos(t)/3,2*sin(u)/3+sin(t)/3,t=0..2*Pi],u=0..2*Pi,axes=none,scaling=constrained,color=blue,frames=50):

> display(m1,m2,m3,m4);

Construct an animated gif file showing the formation of the astroid.

> restart;

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

> m1:=plot([2^(1/2)*(x1-y1)/2,2^(1/2)*(x1+y1)/2,t=0..2*Pi],axes=none,scaling=constrained,color=black):

> x2:=3/4*cos(u)+cos(-3*u)/4:y2:=3/4*sin(u)+sin(-3*u)/4:

> m2:=plot([2^(1/2)*(x2-y2)/2,2^(1/2)*(x2+y2)/2,u=0..2*Pi],axes=none,scaling=constrained,color=green):

> with(plots):

Warning, the name changecoords has been redefined

> x3:=v*3*cos(u)/4+(1-v)*(3*cos(u)/4+cos(-3*u)/4):y3:=v*3*sin(u)/4+(1-v)*(3*sin(u)/4+sin(-3*u)/4):

> m3:=animate([2^(1/2)*(x3-y3)/2,2^(1/2)*(x3+y3)/2,v=0..1],u=0..2*Pi,axes=none,scaling=constrained,color=red,frames=50):

> x4:=3*cos(u)/4+cos(t)/4:y4:=3*sin(u)/4+sin(t)/4:

> m4:=animate([2^(1/2)*(x4-y4)/2,2^(1/2)*(x4+y4)/2,t=0..2*Pi],u=0..2*Pi,axes=none,scaling=constrained,color=blue,frames=50):

> display(m1,m2,m3,m4);

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

> restart;

> m1:=plot([cos(t),sin(t),t=0..2*Pi],axes=none,scaling=constrained,color=black):

> m2:=plot([2*cos(u)-cos(2*u),2*sin(u)-sin(2*u),u=0..2*Pi],axes=none,scaling=constrained,color=green):

> with(plots):

Warning, the name changecoords has been redefined

> m3:=animate([v*2*cos(s*Pi/10)+(1-v)*(2*cos(s*Pi/10)-cos(2*s*Pi/10)),v*2*sin(s*Pi/10)+(1-v)*(2*sin(s*Pi/10)-sin(2*s*Pi/10)),v=0..1],s=1..20,axes=none,scaling=constrained,color=blue,frames=50):

> m4:=animate([2*cos(s*Pi/10)+cos(t),2*sin(s*Pi/10)+sin(t),t=0..2*Pi],s=1..20,axes=none,scaling=constrained,color=blue,frames=50):

> m5:=animate([-cos(2*s*Pi/10)+2*cos(t),-sin(2*s*Pi/10)+2*sin(t),t=0..2*Pi],s=1..20,axes=none,scaling=constrained,color=red,frames=50):

> m6:=animate([-v*cos(2*s*Pi/10)+(1-v)*(2*cos(s*Pi/10)-cos(2*s*Pi/10)),-v*sin(2*s*Pi/10)+(1-v)*(2*sin(s*Pi/10)-sin(2*s*Pi/10)),v=0..1],s=1..20,axes=none,scaling=constrained,color=red,frames=50):

> display(m1,m2,m3,m4,m5,m6);

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

> restart;

> m1:=plot([cos(t),sin(t),t=0..2*Pi],axes=none,scaling=constrained,color=black):

> m2:=plot([3/2*cos(u)-cos(3*u)/2,3/2*sin(u)-sin(3*u)/2,u=0..2*Pi],axes=none,scaling=constrained,color=green):

> with(plots):

Warning, the name changecoords has been redefined

> m3:=animate([v*3*cos(u)/2+(1-v)*(3*cos(u)/2-cos(3*u)/2),v*3*sin(u)/2+(1-v)*(3*sin(u)/2-sin(3*u)/2),v=0..1],u=0..2*Pi,axes=none,scaling=constrained,color=blue,frames=50):

> m4:=animate([3*cos(u)/2+cos(t)/2,3*sin(u)/2+sin(t)/2,t=0..2*Pi],u=0..2*Pi,axes=none,scaling=constrained,color=blue,frames=50):

> m5:=animate([-cos(3*u)/2+3*cos(t)/2,-sin(3*u)/2+3*sin(t)/2,t=0..2*Pi],u=0..2*Pi,axes=none,scaling=constrained,color=red,frames=50):

> m6:=animate([-v*cos(3*u)/2+(1-v)*(3*cos(u)/2-cos(3*u)/2),-v*sin(u)/2+(1-v)*(3*sin(u)/2-sin(3*u)/2),v=0..1],u=0..2*Pi,axes=none,scaling=constrained,color=red,frames=50):

> display(m1,m2,m3,m4,m5,m6);

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

> restart;

> m1:=plot([cos(t),sin(t),t=0..2*Pi],axes=none,scaling=constrained,color=black):

> m2:=plot([2/3*cos(u)+cos(-2*u)/3,2/3*sin(u)+sin(-2*u)/3,u=0..2*Pi],axes=none,scaling=constrained,color=green):

> with(plots):

Warning, the name changecoords has been redefined

> m3:=animate([v*2*cos(u)/3+(1-v)*(2*cos(u)/3+cos(-2*u)/3),v*2*sin(u)/3+(1-v)*(2*sin(u)/3+sin(-2*u)/3),v=0..1],u=0..2*Pi,axes=none,scaling=constrained,color=blue,frames=50):

> m4:=animate([2*cos(u)/3+cos(t)/3,2*sin(u)/3+sin(t)/3,t=0..2*Pi],u=0..2*Pi,axes=none,scaling=constrained,color=blue,frames=50):

> m5:=animate([v*cos(-2*u)/3+(1-v)*(2*cos(u)/3+cos(-2*u)/3),v*sin(-2*u)/3+(1-v)*(2*sin(u)/3+sin(-2*u)/3),v=0..1],u=0..2*Pi,axes=none,scaling=constrained,color=red,frames=50):

> m6:=animate([cos(-2*u)/3+2*cos(t)/3,sin(-2*u)/3+2*sin(t)/3,t=0..2*Pi],u=0..2*Pi,axes=none,scaling=constrained,color=red,frames=50):

> display(m1,m2,m3,m4,m5,m6);

Construct an animated gif file showing the formation of the astroid.

> restart;

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

> m1:=plot([2^(1/2)*(x1-y1)/2,2^(1/2)*(x1+y1)/2,t=0..2*Pi],axes=none,scaling=constrained,color=black):

> x2:=3/4*cos(u)+cos(-3*u)/4:y2:=3/4*sin(u)+sin(-3*u)/4:

> m2:=plot([2^(1/2)*(x2-y2)/2,2^(1/2)*(x2+y2)/2,u=0..2*Pi],axes=none,scaling=constrained,color=green):

> with(plots):

Warning, the name changecoords has been redefined

> x3:=v*3*cos(u)/4+(1-v)*(3*cos(u)/4+cos(-3*u)/4):y3:=v*3*sin(u)/4+(1-v)*(3*sin(u)/4+sin(-3*u)/4):

> m3:=animate([2^(1/2)*(x3-y3)/2,2^(1/2)*(x3+y3)/2,v=0..1],u=0..2*Pi,axes=none,scaling=constrained,color=blue,frames=50):

> x4:=3*cos(u)/4+cos(t)/4:y4:=3*sin(u)/4+sin(t)/4:

> m4:=animate([2^(1/2)*(x4-y4)/2,2^(1/2)*(x4+y4)/2,t=0..2*Pi],u=0..2*Pi,axes=none,scaling=constrained,color=blue,frames=50):

> x5:=v*cos(-3*u)/4+(1-v)*(3*cos(u)/4+cos(-3*u)/4):y5:=v*sin(-3*u)/4+(1-v)*(3*sin(u)/4+sin(-3*u)/4):

> x6:=cos(-3*u)/4+3*cos(t)/4:y6:=sin(-3*u)/4+3*sin(t)/4:

> m5:=animate([2^(1/2)*(x5-y5)/2,2^(1/2)*(x5+y5)/2,v=0..1],u=0..2*Pi,axes=none,scaling=constrained,color=red,frames=50):

> m6:=animate([2^(1/2)*(x6-y6)/2,2^(1/2)*(x6+y6)/2,t=0..2*Pi],u=0..2*Pi,axes=none,scaling=constrained,color=red,frames=50):

> display(m1,m2,m3,m4,m5,m6);

>