Among the definitions of pseudolines listed in my post yesterday, the only one that includes double spirals such as Fermats' spiral is the most general, in which a pseudoline is the image of a line under a homeomorphism of the plane.
One can choose among these definitions according to some sense of mathematical elegance: is it preferable to choose the definition that is stated more simply, or the definition that leads to objects that behave more simply? But here's a stronger reason why I'd like to consider double spirals as pseudolines rather than choosing a definition that rules them out.
In one of my papers on drawing media, some of the results concern a duality relationship between weak pseudoline arrangements and zonotopal tilings. A zonotopal tiling is a partition of the plane into polygonal regions, such that each bounded region is a centrally symmetric polygon. If one starts with a zonotopal tiling with finitely many faces, draws curves that connect pairs of opposite edge midpoints of each bounded face of a zonotopal tiling, and then extends each of these curves to infinity in the unbounded faces without introducing any more crossings, the result is a weak pseudoline arrangement drawn as a planar dual to the original tiling. And conversely, any finite weak pseudoline arrangement is combinatorially equivalent to one formed by this construction.
I would like to be able to replace the part about "finitely many faces" in this by some local finiteness condition that allows for infinitely many tiles. But it's easy to construct locally finite zonotopal tilings that force the dual curves to spiral:
Therefore, I need a definition that allows such spirals to count as pseudolines.