Christian's IJTP paper https://arxiv.org/abs/1211.0784, https://link.springer.com/article/10.1007%2Fs10773-014-2412-2 (paywall) contains a fascinating section on the differential geometry of SU(2) and SO(3), resulting in Figure 3, which is the usual graph of the triangle wave and the negative cosine! But the legend to the figure is: "Fig. 3 Comparison of the geodesic distances on SU(2) and SO(3) as functions of half of the rotation angle ψ. The dashed lines depict the geodesic distances on SO(3), whereas the red curve depicts their horizontal lift to the covering group SU(2)".
This is really interesting. I also think that the relation of this graph with Pearle's model is worth understanding.
It's a strange paper. It has the incredible exploding balls experiment which someone really ought to do. That would lay a lot of ghosts to rest.
Negative distances do not exist, so the legend to the graph isn't quite accurate