Maple17, Windows7, immediately (without specifying the range).

Simply an idea for obtaining an approximate result. This is almost a manual solution plus using the same algorithm to reduce the distance between points on the surface.
It seemed to me that this surface is very sly, and simple techniques work very poorly on it. If, as an initial approximation to the solution, we take the section of the surface by the plane 1.470702751*x1-2.941405502*x2-1.470702751*x3 = 0, which passes through the given points and the middle of the edge, then we get a plane curve (of blue color).
 

By replacing the resulting curve with a finite set of segments, we can slightly change its shape, simultaneously turning this curve into a spatial one. The figure shows a place due to which the overall length of the curve is slightly reduced.

For example, if the plane curve had a length of approximately 3.9762, then with the same number of segments the spatial curve would be approximately 3.9409 (the length of the segment of the plane curve is 0.030000, and the spatial one is  from 0.026501 to 0.029995). All this is rather rough, but the decrease in length occurs with any number of segments (within reason, of course).

Edited 31.07.2024

@dharr   This is a numerical test of the blue curve to be geodesic.
(The text is very amateurish.)
WITH_NORMAL_2.mw

@dharr  Thank you very much for checking the length of the curve. I’ll still have to deal with Ruben’s code, since I’m not very familiar with the theory, and Maple is are still a mystery for me.
As for the blue line, this is also a geodesic. This is the simplest solution, but not the shortest curve between points. That, that this is a geodesic can be verified numerically by constructing osculating planes. The normals to the surface at each point of the curve will lie in the osculating  plane.
For example, for the second curve I did such a selective check.(If I find my code, I'll try to do the same for the blue curve. But I think it's obvious.)

@Rouben Rostamian  Yes, great. 
I've been playing around with  this algorithm.
I got two geodesics. One has a length of approximately 3.8437,



 and the short one has a length of approximately 3.7288. 

@C_R 
Of course, I didn’t dig that deep, but I think that, for example, schoolchildren need to be shown seemingly simple things from different sides.

@Ronan  Thank you. Let me remind you that if you have any questions, you are always welcome.

I remembered the old message regarding degrees of freedom. If you skip the beginning, here is a model of a platform with one degree of freedom. At the same time, it allows you to reproduce quite complex movements, that is, its direct task is uniquely solved by the position of the crank.
Maybe it will come in handy someday.

@Christian Wolinski 
Thanks, I wouldn't have noticed that myself...
I had another idea. For example, if you look separately at the graph of the first equation, you can see that it is intersected by parallel lines (1/2)*Pi^2*x2+x1-(1/2)*Pi = 0, the distance between which is 2*Pi. That is, moving along these straight lines, we track subsets of the curve of the first equation, then move along these sections of the curve and find the intersection with the curve of the second equation. (For the second equation there are also similar straight lines.)

And this way you can also view any selected area.
 

@Christian Wolinski  On behalf of the authors of the publication, I give you a +.

@C_R,  well, I don’t know, I have Maple17, and it gives
only  
for both allvalues(solve(f, x, allsolutions)) and solve(f, x, allsolutions).

This is the equation of a fixed surface
f1 := x3-0.5e-1*exp(x1)/(0.5e-1+x1^4+x2^4);

The equation of a surface that rolls without slipping is shown in the picture.

And a simple torus with radiuses 1 and 0.5.  The picture shows the equation of the surface, rolling without slipping.

To clear my conscience, I added a spline connection to compensate for slight changes in the length of the middle link. This connection is implemented based on the polygon function.


OF_experimental_spline_connection_1_part_3.mw

@mmcdara  So it is, and so it is almost everywhere. I understand you perfectly.

@mmcdara 

 I regret that for many reasons I cannot maintain communication both on general topics and on professional issues. But I can enjoy watching such communication with your participation.
+ from me.