@vv 
Lifetime work Anatoliy Vladimirovich on this subject no. Here is a Russian version in my presentation, where the second part describes the direct Draghilev method.
https://vk.com/doc242471809_437831729 
As for phonetics, then Anatoliy Vladimirovich asked me to write his name in English Draghilev, which is almost exactly corresponds to the pronunciation of the Russian language.

@vv 
Yeah, right, but spell Draghilev. Draghilev method is not quite curve parameterization method in 3d, it was originally a method for solving systems of equations NxN, and it works much better than a continuation methods.
With regard to the parameterization of surfaces, it is the versatility for local application.
Although, let's try your "rotated ellipsoid"

@Carl Love

 I think, that can be simplify the picture by increasing the step from the curves, but then the picture quality will be worse. And the text of the program, probably, can be optimized. How to simplify the algorithm, I do not know.
This, as well as the kinematics of linkages, based on solving underdetermined systems of equations. Many applied problems are related to the presence of free variables, and some of these problems when they clear to me, and when I have enough the degree of ownership of Maple, then the examples of solving these problems I am trying to show.
 

@John Fredsted 
And you did not try to use the break to control or stop the program in proper places? For example, I like this function (break) was very helpful.
 

@Kitonum  It is unnecessary: "numpoints"

    The numerical parameterization to calculate equidistant.  
Surface:
(x1 ^ 2 + x2 ^ 2-0.4) ^ 2 + (x3 + sin (x1 * x2 + x3)) ^ 4-0.1 = 0;

 and equidistant radius R=0.25, surface:
x3-0.25*(sin(4*x1)+sin(3*x2+x3)+sin(2*x2))=0;

@Carl Love 

The text in this topic:

http://www.mapleprimes.com/questions/219995-Finding-A-Convinient-Parametrization-Of-Surfaces#comment233845

         ^{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.