The north and south poles are antipodal points on a sphere and definitions of parallels of latitude and meridians of longitude are well known. A curve that meets every parallel of latitude at the same angle is a loxodrome. Except when the aforementioned angle is a right angle or zero, a loxodrome winds infinitely many times around each pole while approaching it, and has finite length proportional to the cosecant of the aforementioned angle.
Fact: When a loxodrome is stereographically projected onto the plane of the equator with the center of projection at one of the poles, then its image is a logarithmic spiral.
My question is: Where does this fact appear in the literature? Not just a respectable citable source, but also in what variety of contexts does it get mentioned?
(It now occurs to me that the simplest way to prove this fact may be as a corollary of the fact that the stereographic projection is conformal. Earlier I derived it more-or-less by brute force.)