^{Many thanks to all of you for your interest in the idea. The text  is almost the same, as was only to smooth I used polygonplot3d. 
      Carl Love, I'm not sure that the creation of a common procedure would be an easy matter. It is necessary to take into account the direction of parameters on the surface, lengths of parameters , the choice of method for solving differential equations ... More need to connect the transformation that to deal with any kind of smooth surface.
      But, again, for all  local graphs it will be much easier.}

      ^{Maple 17:} 
EXAM_SQ_TORUS_POLYGON.mw
 

Numerical parameterization of 
 (x1 ^ 2 + x2 ^ 2-0.4) ^ 2 + (x3 + sin (x1 * x2 + x3)) ^ 4-0.1 = 0;

@Carl Love 

      In the program for each i  (i = 1..m)  element of La array corresponds to one  array Lg  b dimension. That is, the total number of arrays Lg is m. Together with La they can form a matrix of  mxb size. This is our discrete R^2.  La corresponds to the red curve, and each Lg corresponds to the green curves. (It can be said in another way, how to obtain R^2. This integration intervals [smin (La) .. smax (La)] x [smin (Lg) .. smax (Lg)] for La and Lg, respectively.)
      Locally this parameterization is universal. But for any surface can be an infinite number of numerical parameterization options. As we can see, there may be cases a simple global parameterization.
      For example, a simple global parameterization is obtained for
 (x1 ^ 2 + x2 ^ 2-0.4) ^ 2 + (x3 + sin (x1 * x2 + x3)) ^ 4-0.1 = 0;

@vv 
1. In what is my mistake ?:
        a) My mistake is that the each coordinate of each point of the surface is a function of two variables? Then please justify why it is not.
        b) Or my mistake that you do not like the look of the graph?

2. I do not see  graph of transcendental surface

@vv 

Sorry, I do not speak in English, and I really badly know Maple, maybe that's why I do not understand the purpose of your posts in this thread. You specify me on my mistakes, or (and) you have the method of parameterization of smooth surfaces? Then please:

1. Tell me specifically where is my mistakes
2. Show your method of parameterization of smooth surfaces, for example on the transcendent surface: (x1^2+x2^2-0.4)^2+(x3+sin(x1*x2+x3))^4-0.1=0;
 

     Two variants of the parameterization of one surface:
 (x1 ^ 2 + x2 ^ 2 - 0.4) ^ 2 + (0.9 * x3 + x1 * x2) ^ 4 - 0.1 = 0;

@vv 
inform you: here the two curves. In the animation shows that the curve of one red and the other green.

@vv 
Remove animation from the text, remove  implicitplot3d, make smaller step of curves, and you'll get the graph. Maple builds your exact graph in the same way, only smoothes the image.

@vv 
Any point on the surface can be obtained as a single-valued function of (l1, l2), where l1 and l2 are the lengths of curves. (Within the framework of the numerical solution, of course.) And it has no relation to parametrisation of curves.
I would advise you to first look carefully algorithm, and then draw conclusions.

@vv 
 You have shown the exact solution. I showed the approach to the approximate solution. x, y, z depend on the lengths of the curves on the surface. Curves cover the surface. This is shown in the form of animation, and all that is required in the program.

@vv 
Thank you very much.
But the real problems are almost never associated with the exact solutions, so I do not even think about particular cases.
 I'll show more examples with "torus" surfaces.

@hitstudent 

Example numerical parameterization with a similar surface. If you ever wanted like to parse algorithm, I think that the example will be waiting for you here.
(x1^4+x2^4-2.)^2+x3^4 -1.=0;

EXAM_КВ_БУБЛИК.mw

        The spiral on the surface of "Himmelblau":  z= - (x^2+y - 0.3)^2 - (x+y^2 - 0.7)^2+5;
        Rotates the plane, it intersects the surface. Along the curve of intersection of the plane and the surface is laid distance from the center of rotation depending on the rotation angle. Geodesic is not used.

      Cone: "true" spiral and "untrue" spiral
(program is the same as that for the cylinder).

@vv      However, you have a very well-received "untrue" spiral on the cylinder.
         But what do we do with “bad”' surfaces? For example, as this surface: z = 0.01exp(x)/(0.01+x^4+y^4);