I would not say that using allvalues is necessarily better, unless the user knows exactly what happens in such situations. So, the equation exp(x) = x^2-2 has an infinity of (complex) solutions and solve gives all of them as a RootOf. But allvalues returns only 5 roots as explicit RootOfs.

@Markiyan Hirnyk 

@John Fredsted 

D is a procedure.
D(f):=g  or  D[1](f):=h  fills its remember table.

Maybe, but it's correct, isn't it? After all, it is only a workaround.

@John Fredsted 

I used the argument x only to be able to convert via convert(...,diff), as a workaround.
If you have no arguments, you can use the woekaround:
e := (D[1]@D[1])(f);
unapply(convert(e(x),diff),x);
    (D@@2)(f)

@John Fredsted 

We cannot have everything in this world.

@Markiyan Hirnyk 

@one man 

Unfortunately you use an undefined terminology:

- Equidistant surface (see previous discussions)
- Round parametrization.

So, all we can do is to look at the animations. Do you think that this is enough?

@Markiyan Hirnyk 

It was just a simple ad-hoc approach.

@Markiyan Hirnyk 

But I already did: each point on the surface is moved by fixed distance in the direction of the normal vector.
The code was posted.

chronologically (yours was the second, using the distance to the graph).
Please see that you have considered the graph of sin(x) in the whole real line, not in [0,2*Pi].

Anyway, my impression is that the OP used my version of the definition.

@Markiyan Hirnyk 

@Markiyan Hirnyk 

You took another definition for the equidistance.

If you think that it is better, consider in your example (and notice also that implicitplot did not find all the points):

plots:-implicitplot(['F(s, t)[1] = 2', t = sin(s)], s = -Pi .. 3*Pi, t = -4 .. 4, color = [red, blue], gridrefine = 2)

If you use the first definition:

dist = 1/10 ==>

dist = 1 ==>

Why not, according to the definition considered: each point on the surface is moved by 1/2 units in the direction of the normal vector. I did not check whether the computations made by Maple are correct, have you?

@Markiyan Hirnyk 

And what is not so simple here?

@Markiyan Hirnyk

@one man

It would be nice to say something about the method used in the worksheet.
What "universal parametrization" have you used? Otherwise, the reader must use a "reverse engineering"  technique to find out.

If the surface is already parametrized, one may use something like this:

CP:=(v,w)->[v[2]*w[3]-v[3]*w[2], v[3]*w[1]-v[1]*w[3], v[1]*w[2]-v[2]*w[1]]: #Cross Product
IP:=(v,w)->v[1]*w[1]+v[2]*w[2]+v[3]*w[3]: # Inner Product
UV:=v -> v/~sqrt( IP(v,v)); #Unit Vector

r:=[s*cos(t), s*sin(t), s^2]:          # Example, paraboloid (parametric)
Nr:=UV( CP( diff(r,s), diff(r,t) ) ): #Unit Normal vector to r
Er:=simplify(  r + 1/2*Nr) ;           # Equidistant to r


p1:=plot3d( r, s=1 .. 2, t=0..2*Pi ):  p2:=plot3d( Er, s=1 .. 2, t=0..2*Pi ):
plots[display](p1,p2);






Edit. OP's animation using this method:

The problem is that Maple gives the result without assumptions, even if for a complex k, the integral may not exist (converge) e.g. for k = I.

@m3rcur1al 

You may use the Explore command and play with the parameters, as in the example:

Explore(fsolve(c1*exp(x)-c2*sin(x), x), parameters=[c1=1..15, c2=-3..13]);

You may prefer to replace fsolve with a plot.