O15-056
> restart;with(plots):
> crbe:=a->[(cos(t))^3+a*sin(t),(sin(t))^3+a*cos(t),t=-Pi..Pi];

  crbe :=

                    3                   3
        a -> [cos(t)  + a sin(t), sin(t)  + a cos(t), t = -Pi .. Pi]

> des:=plot([crbe(-2),crbe(-3/2),crbe(-1),crbe(0),crbe(1),crbe(3/2),crbe(2)]):display(des);# t->Pi/2-t donne la sym/y=x ; t->t+Pi donne la sym/O

> deriv:=map(diff,[(cos(t))^3+a*sin(t),(sin(t))^3+a*cos(t)],t);# Points singuliers : a=3*cos(t)*sin(t) ie 2*a/3=sin(2*t) pour abs(a)<=3/2

  deriv :=

                  2                            2
        [-3 cos(t)  sin(t) + a cos(t), 3 sin(t)  cos(t) - a sin(t)]

> tt:=(arcsin(2*a/3))/2;sing1:=[(cos(tt))^3+a*sin(tt),(sin(tt))^3+a*cos(tt),a=-3/2..3/2];sing2:=[-(cos(tt))^3-a*sin(tt),-(sin(tt))^3-a*cos(tt),a=-3/2..3/2];sing3:=[(sin(tt))^3+a*cos(tt),(cos(tt))^3+a*sin(tt),a=-3/2..3/2];sing4:=[-(sin(tt))^3-a*cos(tt),-(cos(tt))^3-a*sin(tt),a=-3/2..3/2];

                       tt := 1/2 arcsin(2/3 a)


  sing1 :=

                3                     3
        [cos(%1)  + a sin(%1), sin(%1)  + a cos(%1), a = -3/2 .. 3/2]

  %1 := 1/2 arcsin(2/3 a)


  sing2 := [

                3                      3
        -cos(%1)  - a sin(%1), -sin(%1)  - a cos(%1), a = -3/2 .. 3/2

        ]

  %1 := 1/2 arcsin(2/3 a)


  sing3 :=

                3                     3
        [sin(%1)  + a cos(%1), cos(%1)  + a sin(%1), a = -3/2 .. 3/2]

  %1 := 1/2 arcsin(2/3 a)


  sing4 := [

                3                      3
        -sin(%1)  - a cos(%1), -cos(%1)  - a sin(%1), a = -3/2 .. 3/2

        ]

  %1 := 1/2 arcsin(2/3 a)

> des2:=plot([sing1,sing2,sing3,sing4],color=black):display(des2);

> display([des,des2]);

>