@Ronan  Thank you for your interest to this topic.
Want to draw your attention that for my part there is practically no reaction to the replicas of a certain person (I will not mention his name). For myself, I have long concluded that it is useless to communicate with him for many reasons, or rather, even harmful to one's own reputation and health.

@fereydoon_shekofte  Thank you for your kind words.
When I had a job, I worked on developing algorithms, but I was not able to implement many ideas. A few years ago I met Maple, and Maple inspired me to return to those tasks.
I would be happy if this was useful for someone.

@kuwait1 
Do you understand that the graphs depict the equations of the first derivatives of f (\ epsilon, \ phi)?
You do not show me the f (\ epsilon, \ phi), and I myself integrated the partial derivatives and looked at the plot of
the f (\ epsilon, \ phi) with help of plot3d. And you yourself can see that the minimum of f (\ epsilon, \ phi) in this definition area is an infinite flat set.

@kuwait1 

Sorry, I still do not see your original function f (\ epsilon, \ phi).
As for your last equations, I took them from fsolve, and denoted x and y.
On an implicit graph x, it is clear that \ epsilon is practically 0, and the graph of y is empty.

I think you still have these options:
1) to look for min ((f (\ epsilon, \ phi)) ^ 2). It's working with the function itself without derivatives.
2) scale the variables "\ epsilon" and "\ phi" so that the range "\ phi" is much wider for fsolve.
equation_solve(2).mw

@kuwait1 Ah, here's the thing. And where is your f (\ epsilon, \ phi)?

By the way, it may well be that your min has an infinite number of solutions.

@kuwait1  You really have only one equation: either x or y. There are many ways to solve one equation with several variables. First try the easiest way: set the desired values of the epsilon and  then solve one equation for another variable using fsolve.

The graphs show that your equations are equivalent:
with(plots, implicitplot):
implicitplot(x, `ε` = -5 .. 5, phi = -5 .. 5, numpoints = 20000, color = red);
implicitplot(y, `ε` = -5 .. 5, phi = -5 .. 5, numpoints = 20000, color = blue);
implicitplot([x, y], `ε` = -5 .. 5, phi = -5 .. 5, numpoints = 20000, color = [red,blue]);
or
x-y;
0;

 

Additional curling of a Möbius strip (rolling without slipping).
Mobius_strip_rolling_Additional_curling.mw

@mathstutor  And what about GeoGebra?
https://en.wikipedia.org/wiki/GeoGebra

@tomleslie I also read and can not understand why?

Addition to the topic:
http://www.mapleprimes.com/posts/202821-Calculating-Link-Mechanisms?submit=comment

Velocity and acceleration along the trajectories:
https://vk.com/doc242471809_439567938
https://vk.com/doc242471809_439567889

Now in the Application Center. ( More detailed description of some examples.)
http://www.maplesoft.com/applications/view.aspx?SID=154228  

Interestingly, and what specific opportunities has MapleSim versus Maple?
For example, in solving this problem:
http://www.mapleprimes.com/posts/204684-Linkage--Mechanisms-#comment201753

On the projected curve are only two points, h=0.01
The distance from the point to the surface is the shortest 
curve_between_the_surfaces_H.mw

For this curve all right because of the distance. This segment connects the points of the projection. The projection is interrupted by analogy with equidistant.