@Markiyan Hirnyk In order to you were not bored, try something like this. As a hint showing the surface equation and remind you that the motion on the surface in any space can be not only a picture, but at the same time, the solution of the corresponding equation.
x3-0.1e-1*exp(x1)/(0.94e-2+x1^4+x2^4)=0;  
x3-.1*(sin(4*x1)+sin(3*x2+x3)+sin(2*x2))=0;


https://vk.com/doc242471809_361089459
https://vk.com/doc242471809_360840447

Additional examples  on other surfaces

     Your system is one equation for two variables x1, x2:

eq: = tanh (x1) + cosh (x1 + x2) - 2.

     Any solution to this system is a solution of the original system. Choose from all what you need.

o@Axel Vogt 

       Consider the simplest situation. Numerical method in real space. Points of the curve are substituted in the plane equation P (x) = 0. If abs (P (x)) <= delt, the point belongs to a plane. The equation of the plane is made by the first three points of curve that allow you to obtain the equation of the plane. Accordingly, if N (the dimension of the space R ^N) > 3, then for the equation of the plane should be taken  > 3 points of the curve.  The curve can be set or separate equations coordinates of the points of the curve, or by solving a system of nonlinear equations N-1 to the number of variables N, polynomial or transcendental equations.
       Sorry, my English through a Google translator.

@Markiyan Hirnyk 

      For all points of the curve is considered one single equation of the plane. "Any three points" - are any suitable first three points on the curve.

Can use any other suitable method.

@Markiyan Hirnyk 

      Hirnyk, my best friend. Let's proceed as follows. Since it is necessary to compare the comparable, make yourself at least something on the proposed directions. Show that you can to do it. And Use package Sergei Moiseyev I can too, but do it only if necessary. You were asked at forum Exponenta.ru to learn how to solve the ODE with boundary conditions by using package of Sergei Moiseyev . You pretended that not to notice the proposal. And that now rend the air?
      Yes, relax and learn to write real programs.

@Markiyan Hirnyk 

      The number of points in my example m: = 748, for a few seconds calculations. Points can be many times greater, and they are sequentially forming a curve. If desired, can be positioned at a point equidistant from each other along the curve. You have neither the one nor the other and.
      You, my friend Hirnyk, all also continue to engage in this nonsense with a perseverance worthy of better application. Relax. Do not torture post.

@Markiyan Hirnyk 

I see you have a constant need anything somewhere to write. Please justify your words, adhering to the text of my posts, or acknowledge his complete incompetence in this matter. (Although the latter is so obvious.)

D_Method_12d.mw

      Draghilev’s method and 12d curve.  We solve the system of 11 polynomial equations with 12 variables.  This system has infinitely many solutions. One of the subsets of the set of solutions of this system is given curve in the space 12d. For visualization of the curve, we combine arbitrarily xi (i = 1..12) by three, and show them all together like a movie in our 3d. Any point on the curve may be easily checked whether it is solution of the system.
      Someone said  "A", says "B".

@Markiyan Hirnyk 

Edited the first message. It was showed what happens to the original line while minimizing the distance between the points. Other algorithms operate  on the basis of moving of points on the surface too. Movement of points occurs according to a particular task.

The method of calculating of spatial linkage.  Here in the Application Center

(If the link itself does not work, then you can use the clipboard and paste it into the address bar of the browser)
https://www.maplesoft.com/applications/view.aspx?SID=154228

Geometry, velocity and acceleration.

https://vk.com/doc242471809_321893668
https://vk.com/doc242471809_321893668
https://vk.com/doc242471809_324249241

@Carl Love 

This is not the specific device. This is variant of method of calculating the geometry and kinematics of "arrow" by  Maple  tools   for any possible movement of the body in 3d, when its surface is described by a polynomial equation.