Editing Spiral (section)

=== Spherical spirals ===
[[File:Kugel-spirale-1-2.svg|thumb|upright=1.2|Clelia curve with <math>c=8</math>]]

Any [[cylindrical map projection]] can be used as the basis for a '''spherical spiral''': draw a straight line on the map and find its inverse projection on the sphere, a kind of [[spherical curve]].

One of the most basic families of spherical spirals is the [[Clelia curve]]s, which project to straight lines on an [[equirectangular projection]]. These are curves for which [[longitude]] and [[colatitude]] are in a linear relationship, analogous to Archimedean spirals in the plane; under the [[azimuthal equidistant projection]] a Clelia curve projects to a planar Archimedean spiral.

If one represents a unit sphere by [[spherical coordinates]]

: <math>
x = \sin \theta \, \cos \varphi, \quad
y = \sin \theta \, \sin \varphi, \quad
z = \cos \theta,
</math>

then setting the linear dependency <math> \varphi=c\theta</math> for the angle coordinates gives a [[parametric curve]] in terms of parameter {{tmath|\theta}},<ref>Kuno Fladt: ''Analytische Geometrie spezieller Flächen und Raumkurven'', Springer-Verlag, 2013, {{ISBN|3322853659}}, 9783322853653, S. 132</ref>

:<math>
\bigl( \sin \theta\, \cos c\theta,\, \sin \theta\, \sin c\theta,\, \cos \theta \,\bigr).
</math>

<gallery>
KUGSPI-5 Archimedische Kugelspirale.gif|Clelia curve
KUGSPI-9_Loxodrome.gif|Loxodrome
</gallery>

Another family of spherical spirals is the [[rhumb line]]s or loxodromes, that project to straight lines on the [[Mercator projection]]. These are the trajectories traced by a ship traveling with constant [[bearing (navigation)|bearing]]. Any loxodrome (except for the meridians and parallels) spirals infinitely around either pole, closer and closer each time, unlike a Clelia curve which maintains uniform spacing in colatitude. Under [[stereographic projection]], a loxodrome projects to a logarithmic spiral in the plane.