Degenerate diffusions on the special Euclidean group of the plane arise in a number of applications in filtering theory such as the construction of dead-reckoning priors in nonholonomic mobile robot pose estimation, and in inpainting and stochastic completion in the study of visual perception. Two very different solution methodologies have been pursued in the literature: (1) Gaussian distributions in exponential coordinates for very concentrated probability densities; (2) noncommutative Fourier expansions for very distributed probability densities. Here we compare and contrast these methodologies and examine the range over which they provide comparable answers and quantitatively analyze when one should be used over the other.