@Jarekkk 

thanks at all Jarekkk, I am grateful to you

@Jarekkk 

When i use 

M := 10;
f :=i -> 2*(i-1)*Pi/M :       f(M+1) := f(1):        v := Vector(M+1, f);

x := cos~(v)/~(1+~ k*cos~(2*v));              y:= sin~(v)/~(1+~ k*cos~(2*v));  

x1dot := diff~(x, v);                                   y1dot :=diff~(y, v);

x2dot := diff~(x, v$2));                              y2dot := diff~(y, v$2));

it does not work numerically.

But when i use f :=i ->theta[i]    instead of     f :=i -> 2*(i-1)*Pi/M   it works well

@Jarekkk 

When i use 

M := 10;
f :=i -> 2*(i-1)*Pi/M :       f(M+1) := f(1):        v := Vector(M+1, f);

x := cos~(v)/~(1+~ k*cos~(2*v));              y:= sin~(v)/~(1+~ k*cos~(2*v));  

x1dot := diff~(x, v);                                   y1dot :=diff~(y, v);

x2dot := diff~(x, v$2));                              y2dot := diff~(y, v$2));

it does not work numerically.

But when i use f :=i ->theta[i]    instead of     f :=i -> 2*(i-1)*Pi/M   it works well

@Jarekkk    thanks Jarekkk

but when i want the program do the your steps numerically    i found an error?

M := 10;
f :=i -> 2*(i-1)*Pi/M :       f(M+1) := f(1):        v := Vector(M+1, f);

x := cos~(v)/~(1+~ k*cos~(2*v));              y:= sin~(v)/~(1+~ k*cos~(2*v));  

x1dot := diff~(x, v);                                   y1dot :=diff~(y, v);

x2dot := diff~(x, v$2));                              y2dot := diff~(y, v$2));

instead by file

@Jarekkk    thanks Jarekkk

but when i want the program do the your steps numerically    i found an error?

M := 10;
f :=i -> 2*(i-1)*Pi/M :       f(M+1) := f(1):        v := Vector(M+1, f);

x := cos~(v)/~(1+~ k*cos~(2*v));              y:= sin~(v)/~(1+~ k*cos~(2*v));  

x1dot := diff~(x, v);                                   y1dot :=diff~(y, v);

x2dot := diff~(x, v$2));                              y2dot := diff~(y, v$2));

instead by file

If      x:= cos(θ[i])/(1+cos(θ[i]);     y[i] := sin(θ[i])/((1+cos(θ[i])

what can i do 

If      x:= cos(θ[i])/(1+cos(θ[i]);     y[i] := sin(θ[i])/((1+cos(θ[i])

what can i do 

thanks, but 

θ[1] := 0; for q to M+1 do θ[q+1] := θ[q]+dθ; end do     dθ=2Pi/M,   M:=100

when i use this program i found an error,

thanks, but 

θ[1] := 0; for q to M+1 do θ[q+1] := θ[q]+dθ; end do     dθ=2Pi/M,   M:=100

when i use this program i found an error,

I mean

θ[1] := 0; for q to M+1 do θ[q+1] := θ[q]+dθ; end do     dθ=2Pi/M,   M:=100

I mean

θ[1] := 0; for q to M+1 do θ[q+1] := θ[q]+dθ; end do     dθ=2Pi/M,   M:=100

@Ninetrees 

Yes of course.
We are all here to learn.
I hope you will finish  your Ph.D in short time.
With best wishes.
Dr. A. R. El Dhaba. Mathematics Department - faculty of Science - Damanhour University

@Ninetrees 

Yes of course.
We are all here to learn.
I hope you will finish  your Ph.D in short time.
With best wishes.
Dr. A. R. El Dhaba. Mathematics Department - faculty of Science - Damanhour University

@Ninetrees

Sir Don't puzzle.  follow me right click on the bule (Ninetrees) and chose open Link in new window or open  link in new tab you will find your profile (4 years, 295 days) which equal=1755 days.
and from my profile (0 years, 147 days) =147 days

the ratio = 1755/147=585:49.

Until now I couldn't find my answer for my question.

@Ninetrees

Sir Don't puzzle.  follow me right click on the bule (Ninetrees) and chose open Link in new window or open  link in new tab you will find your profile (4 years, 295 days) which equal=1755 days.
and from my profile (0 years, 147 days) =147 days

the ratio = 1755/147=585:49.

Until now I couldn't find my answer for my question.