7-8 The cycloid:(t+sint,cost)

> restart;

> x:=t+sin(t):

> y:=cos(t):

> a:=plot([x,y,t=0..7*Pi],color=red,scaling=constrained,axes=none):

> a;

> x1:=diff(x,t):

>

> y1:=diff(y,t):

> x2:=diff(x1,t):

> y2:=diff(y1,t):

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

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

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

> with(plots):

Warning, the name changecoords has been redefined

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

> b;

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

> c;

> d:=animate([(1-s)*xc+s*x,(1-s)*yc+s*y,s=0..1],t=0..2*Pi,color=black,scaling=constrained,axes=none):

> d;

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

>

>