All statistical methods/approaches/tools that I have encountered so far are always in Cartesian coordinates.
Why not polar? Doesn't math "work out" a little different in polar coordinates? (for example drawing a spline is easy $r = \theta $)
Couldn't that lead to some interesting results?
Best Answer
I'm not sure what is the question here. I'll assume that it's "why most statistical results are expressed in Cartesian coordinates?"
I think it's because the most common applications of statistics are in non-physical sciences and business where the symmetry is not clear or obvious. In physical sciences the symmetry is very important and very well studied. For instance, consider particle scattering such the most basic one taught in high school, the Rutherford scattering.
In a nutshell, you have two particles colliding. So, you can write the equations of motion in terms of one particle moving in the central potential located in the center of masses. The central potential has spherical symmetry. That's why it's more natural (call it easy) to write the motion equations not in Cartesian but in spherical coordinates. You end up computing the probability distribution of scattering into a unit area at a certain angle in spherical coordinates. It would have been very cumbersome to do this in Cartesian coordinates.
In quantum mechanics the wave function is related to the probabilities of states. The non-cartesian coordinates are used heavily. It's all about the symmetry of your phenomenon, which is a very big deal in physics. For instance, look at this lecture on quantum scattering. You can see how the presentation switches from the density function in spherical coordinates to Cartesian and back when they look at central potential and a square well. The former has a spherical symmetry, while the latter doesn't.
Another example is a rainbow angle. If you think about why the rainbow is a perfect circle, it's an interesting and simple phenomenon. The symmetry is formed around an axis going through the Sun and the rain area relative to where you stand. The Sun rays come in parallel, i.e. the intensity is a uniform distribution in the area perpendicular to the direction of sun lights. The rain droplet is a sphere, when the Sun light gets out of the droplet its intensity is not uniform anymore! Its distribution is symmetrical though, and you can compute the density function in angular coordinates. It'll have a peak at a rainbow angle, which is why they show up as arcs to an observer. This also can be seen as an example of the probability distribution transformation from uniform to something else.
So, using non-Cartesian coordinates is not a matter of being fancy, but a simple matter of convenience. Physicists use whatever math works best in any given situation.
In non-physical sciences researchers pay little attention to symmetry. I think the reason is that the symmetries are not obvious or maybe not present at all. They're not a factor in most cases. An alternative hypothesis would be to suspect that social scientists are not trained to spot and exploit the symmetries of phenomena, but that seems unlikely to me.
So, I'll finish with an example of usage of non-Cartesian coordinates in everyday statistics, which is picking random points on a sphere. This is a common problem in non-physical sciences too.