You're referring to a transformation from a pair of independent variates $(X,Y)$ to the polar representation $(R,\theta)$ (radius and angle), and then looking at the marginal distribution of $\theta$.

Note that if you scale the two variables, X and Y by some common scale (e.g. go from U(-1,1) to U(-10,10) or from N(0,1) to N(0,20) on both variables at the same time) that makes no difference to the distribution of the angle (it only affects the scale of the distribution of the radius). So let's just consider the unit cases.

First consider what's going on with the uniform case. Specifically, look at the density associated with an element of angle, $d\theta$ near the horizontal (near angle $\theta=0$) and on the diagonal (near angle $\theta=\pi/4$):

[![enter image description here][1]][1]

Clearly the probability element $df_\theta$ (i.e. area) corresponding to an element of angle ($d\theta$) is larger when the angle is near one of the diagonals. This completely accounts for the pattern you see.

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By contrast, if the joint distribution is rationally symmetric about the origin then the probability element at some angle doesn't depend on the angle (this is essentially a tautology!). The bivariate distribution of two independent standard Gaussians is rotationally symmetric about the origin.


  [1]: https://i.sstatic.net/cdvFV.png