@Thomas Richard  Thank you very much for such a detailed answer. It seemed better to continue only with Maple.
(As for my Maple drawing(s), of course, it's not modeling, it's simply a visualization of a calculations.)

@Bendesarts Could you briefly tell about the problem in connection with the decision which arose such expressions?

More versions of cam mechanisms

  @tomleslie Example with Isolate: all solutions of a system of polynomial equations.

example_with_isolate.mw 

@tomleslie Root Finding [Isolate] works with Gröbner bases and thus find all the solutions of a polynomial system (when replacing variables for the sum of the real and the imaginary part find all real and all complex solutions. In Mathematica it seems, Nsolve).  On this basis, the system can be examined for compatibility  and an infinite number of solutions. But a polynomial systems only.

For another type of systems is a very powerful tool is DirectSearch [SolveEquations].

If there is a continuous infinite set of solutions recommend Draghilev's method.

@tomleslie  Excuse me, I do not know English, but I realized that this is not a solution, and to the answer to question Muhammad Usman has nothing to do.

tomleslie, check your solution on the value of the discrepancies

Surface type Himmelblau and cylinder

https://vk.com/doc242471809_437007771 
 
 
(x1^2 + x2 - 0.3)^2 + (x1 + x2^2 - 0.7)^2 + x3 - 5 = 0.

 @roya2001 I almost forgot about the intersectplot. As you can see, the other answers in real space are not available.

2DOF_2.mw  

@roya2001 

For example, You can check this by using Direct search optimization package.

http://www.maplesoft.com/applications/view.aspx?SID=87637

One more example with animation

@Bendesarts I do not know how to use Maplesim. Is it impossible to use in Maplesim equations?

I think that the best and universal parameterization would parameterization based on geodesic curves. It is only necessary to have time to do it.
  

        This is the easiest program text of the method, which I could do. I think someone wants to deal with the idea of the method, the text will help him and it will be easier to deal with other examples of the application and then create their own programs.

@Axel Vogt       
     Thank you for your attention to the subject and for your constructive, as I understood  the message. If Google translator has translated all right on both sides, then:
     Yes, 2d and 3d convenient to use implicitplot  to see, but what is the accuracy and computation time? And how to make the calculation of motion mechanisms when the dimension, for example, > 12d?  As for the abstract shapes in the form of a complex set of solutions and higher dimensions, we can move in very small increments (h) and to monitor the value of abs (F (X)) <eps, and create a set of points, constantly changing direction, or to pick up auxiliary equation. When we seek solutions by this method, it is sufficient to have a single point on the set of solutions that from it very quickly and accurately obtain an infinite number of other points. We do not iterate over all grid points in some region of space where the supposed existence of a solution but move on to the set of solutions.
     My example  "bublik" is just an illustration, but examples the mechanisms it is work that is very difficult to perform using implicitplot (or rather impossible), but using this method mechanisms are calculated quickly and easily.  
     In my posts given examples of geodesic computations. Geodesics calculated numerically in any direction and at any distance on any piecewise smooth surfaces. There, too, the proposed method is applied.