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.)


1 Answer 1



In the above Mathographics book is given a beautiful construction of chess-board projected on a sphere while discussing stereographic projection and inversions .. that I can never forget. It should be noted that in inversions the angle $\alpha $ between curve and meridian is conserved conformally, but in the opposite sense of direction. In the above limited google preview page 73 is not included.

The topic is included in several text-books of differential geometry. Loxodromes can be derived using geometry of differential triangles on sphere radius $a$ with cylindrical coordinates thus:

$$ \cos\phi = r/a , dr/ \sin \phi = r\cdot d \theta \cot\alpha $$

$$ \cot \alpha \cdot \theta = \int \frac {dr}{ r \sqrt{1- (r/a)^2)}} $$

leading to:

$$ r/a = \operatorname{sech} (\theta\cdot \cot \alpha ) , z/a = \tanh (\theta \cdot \cot \alpha ) $$

IIRC, The Sphere / Bowl of Inversion / Tangent plane at South Pole project (orthogonal set of) Loxodromes / Logarathmic Spirals/ Checker (chess ) board respectively from North Pole.

Lectures on Classical Differential Geometry 2nd Edition DJ Struik Pp 174-175 for classical stereographic projections.Loxodromes on page 60, mapping to straight lines of chessboard as Mercator projection mentioned on page 173 last line.


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