NLPSolve () does not work perfectly (does not always work), but it can be used for real problems.
The projection of the curve on the surface of formally constructed correctly.
curve_between_the_surfaces_H.mw

Projection of curve from one surface (green) to another. Projection made by the normal to the second surface.
curve_between_the_surfaces_1.mw

@Bendesarts The first link to the text, where the second part has an authentic description of the method.

https://vk.com/doc242471809_437831729
http://old.exponenta.ru/educat/systemat/selitskiy-ivanov/index.asp 
http://www.mapleprimes.com/posts/204684-Lever-Mechanisms-

I thought it was all known here. The first two links to Russian. The third link is what was promised to you here in the topic:

http://www.mapleprimes.com/posts/202821-Calculating-Linkage-Mechanisms#comment201716

@Bendesarts  The theory is very simple. This Cauchy problem for ODE and Cramer's rule for solving systems of linear equations. (The homogeneous linear equations with free variables.)
Everything else belongs to the application areas where the possible to use  the method.
I hope I understand your question?

@fereydoon_shekofte  It is not I, but Draghilev method.

@tontu  There are three equations x = f1 (t1, t2, t3), y = f2 (t1, t2, t3), z = f3 (t1, t2, t3), and six variables x, y, z, t1, t2, t3, that is, we have  space R^6. Choose any convenient to you the projections of  R^6 on R^3 and look at their graphs.

        If in the text to write "#" in front of the approximate solution x0] 1], x0 [2], and remove the "#" from any version of Optimization package then no one variant will not work.
OPT_DIF.mw 

@Kitonum  Any quadrilateral and so on is divided into triangles

@vv  No words

@Kitonum  Thanks, but, yes, unfortunately, this only applies to the simplest cases.

     Can anybody explain to a "local specialist', that he had to teach math to understand that in the denominator of the parameter should not be? This smooth curves in 3d on the all set of definitions.

    This parametric curve for second-order surface and for plane in the common form equations.
   

         PLAN_CURVE_3d_COMMON.mw.

@tomleslie  I remain in the dark about whether I was able to answer your question: "Purpose ???"

For those who have not used the program text. This parametric equations of the curves in the second and third Figs respectively.

 x1^2+0.1*x2^2+x3^2-9=0;
 x1+3*x3+1=0;
x1(s)= -(135121896351/50000000000)cos(2s)+(8407313781/10000000000)sin(2s)-4999999999/50000000000;
x2(s)= -(45040632117/5000000000)sin(2s)-(2802437927/1000000000)cos(2s);
x3(s)= (45040632117/50000000000)cos(2s)-(2802437927/10000000000)sin(2s)-14999999997/50000000000;
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x1^2-0.1*x2^2+x3^2-9=0;
x1+3*x3+1=0;
x1(s)= -(390000000009/200000000000)exp(2s)-(205384615389/200000000000)exp(-2s)-10000000001/100000000000;
x2(s)= -(130000000003/20000000000)exp(2s)+(68461538463/20000000000)exp(-2s);
x3(s)= (130000000003/200000000000)exp(2s)+(68461538463/200000000000)exp(-2s)-30000000003/100000000000;

@tomleslie   Here is the purpose: parametric equation of second-order curve in 3d.
For example, the first curve equations:
x1(s)= -135000000013/5000000000+(166055512773/10000000000)exp(s)+(101773194523/10000000000)exp(-s);
x2(s)= (7828707271/10000000000)exp(-s)-10000000001/10000000000;
x3(s)=-(166055512773/10000000000)exp(s)-(54800950897/5000000000)exp(-s)+140000000013/5000000000;
And where are your equations? Read carefully, please...