@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.
 

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