# How is this a gauge choice mathematically?

I've been reading an article about the "square cat", which is described as the system bellow

Such system is a deformable body that can change $$a$$ and $$\theta$$ but has $$b$$ fixed. The article uses this to model a cat falling without the action of external forces and tries to show if the cat can change it's orientation.

The idea basically is to represent as in this figure the orientation by the angle $$\Phi$$ between the $$x$$-axis and the $$b$$ axis of the body. Now comes the thing: the problem can be stated as a gauge theory problem considering the space of shapes $$S$$ of the body as the space of tuples $$(a,\theta,e^{i\phi})$$ with $$e^{i\phi}\in U(1)$$ representing the orientation by an angle $$\phi$$.

Then we have also the space of unoriented shapes that is $$S/U(1)$$ where we identify shapes which can be obtained one from another by just reorienting them. This space can be identified with the space of pairs $$(a,\theta)$$. Being then $$\pi : S\to S/U(1)$$ given by $$\pi(a,\theta,e^{i\phi}) = (a,\theta)$$ we have $$(S,\pi, S/U(1))$$ a principal bundle with structure group $$U(1)$$.

That is all fine, my doubt is the following: as I've read sometimes, a choice of gauge is a choice of local trivialization for the bundle, which is equivalent to a choice of a section of the bundle. That is a function $$\varphi : S/U(1)\to S$$. My understanding is: such a function gives for each unoriented shape one default oriented shape so that we can measure the orientation from there.

Now, the article, on the other hand says differently. It says that a choice of gauge is a choice of how we measure $$\Phi$$. In that case, one gauge choice would be measure $$\Phi$$ as the angle between the $$x$$-axis and the $$b$$ axis and another gauge choid would be measure $$\Phi$$ as the angle between the $$x$$-axis and the line connecting two opposing masses.

I see the following difference between my understanding and the article: on my understanding, a choice of gauge would be to choose the initial angle with the $$x$$-axis. That is $$\varphi(a,\theta) = (a,\theta,1)$$ we measure orientation with respect to the $$x$$-axis and $$\varphi(a,\theta) = (a,\theta, e^{i\phi})$$ we measure orientation with respect to the line with angle $$\phi$$ with the $$x$$-axis.

On the article, a choice of gauge would be to choose the other line. Measure the orientation with respect to the $$b$$-axis or with respect to the line joining oposing masses. I can't see how this idea from the article is a choice of gauge. How can this b written as a section of the bundle? A section, as it seems, would give simply the initial angle with the $$x$$-axis rather than determine where to stop measure the angle.

Choosing a gauge usually means implicitly performing a gauge transformation such that a given condition holds, such as choosing a gauge $\partial_\mu A^\mu = 0$ in electrodynamics.
A local trivialization of a $G$-principal bundle $G \to P \overset{\pi}{\to} M$ is given by an open covering $\{U_i\}$ of $M$ and diffeomorphisms $\{\phi_i : U_i \times G \to \pi^{-1}(U_i)\}$ such that $\pi \circ \phi_i = \mathrm{id}_M$. It is called that way since $M \times G$ would be the trivial bundle, and such a local trivialisation makes the condition that any bundle is locally isomorphic to a trivial bundle explicit.
These two concepts are not exactly the same, but "choosing a gauge" is particularly simple if you give it on the local trivialisations. Any local trivialisation looks like $(x,g) \mapsto (x,g t(x))$ with $t : U_i \to G$ a smooth function. This local trivialisation is a diffeomorphism since multiplication with a group element is a diffeomorphism of the Lie group. A choice of gauge is now simply fixing $t_i : U_i \to G$ for every $U_i$ in a way that is compatible on the overlaps $U_i \cap U_j$, but since your bundle is trivial, this condition need not concern us.
Now, the "gauge choice of $\Phi$" in your question amounts to choosing a function $\chi : S/\mathrm{U}(1) \to \mathbb{R} \cong \mathfrak{u}(1)$ that assigns to every pair $(a,\theta)$ the angle $\chi(a,\theta)$ between the $b$-axis and the axis joining two opposite masses for that value of $a$ and $\theta$. The tuple $(a,\theta,\mathrm{e}^{\mathrm{i}\Phi})$ where $\Phi$ was measured w.r.t. to the angle between the $x$-axis and the $b$-axis, is then sent to $(a,\theta,\mathrm{e}^{\mathrm{i}\Phi + \chi(a,\theta)})$, so the gauge transformation is given by
$$t : S/\mathrm{U}(1) \to \mathrm{U}(1), (a,\theta) \mapsto \mathrm{e}^{\mathrm{i}\chi(a,\theta)}$$