Editing Spiral (section)

== Three-dimensional ==
=== Helices {{anchor|Spiral or helix}} ===
[[File:Schraube und archimedische Spirale.png|right|thumb|An Archimedean spiral (black), a helix (green), and a conical spiral (red)]]
Two major definitions of "spiral" in the [[American Heritage Dictionary]] are:<ref name=free>"[http://www.thefreedictionary.com/spiral Spiral], ''American Heritage Dictionary of the English Language'', Houghton Mifflin Company, Fourth Edition, 2009.</ref>
# a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.
# a three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a [[helix]].

The first definition describes a [[Plane (mathematics)|planar]] curve, that extends in both of the perpendicular directions within its plane; the groove on one side of a [[gramophone record]] closely approximates a plane spiral (and it is by the finite width and depth of the groove, but ''not'' by the wider spacing between than within tracks, that it falls short of being a perfect example); note that successive loops ''differ'' in diameter. In another example, the "center lines" of the arms of a [[spiral galaxy]] trace [[logarithmic spiral]]s.

The second definition includes two kinds of 3-dimensional relatives of spirals:
* A conical or [[volute spring]] (including the spring used to hold and make contact with the negative terminals of AA or AAA batteries in a [[Battery (electricity)|battery box]]), and the [[vortex]] that is created when water is draining in a sink is often described as a spiral, or as a [[conical helix]].
* Quite explicitly, definition 2 also includes a cylindrical coil spring and a strand of [[DNA]], both of which are fairly helical, so that "helix" is a more ''useful'' description than "spiral" for each of them. In general, "spiral" is seldom applied if successive "loops" of a curve have the same diameter.<ref name=free/>

In the side picture, the black curve at the bottom is an [[Archimedean spiral]], while the green curve is a helix. The curve shown in red is a conical spiral.

{{redirect|Space spiral|the building|Space Spiral}}

Two well-known spiral [[space curve]]s are ''conical spirals'' and ''spherical spirals'', defined below.
Another instance of space spirals is the ''toroidal spiral''.<ref name="von Seggern 1994 p. 241">{{cite book | last=von Seggern | first=D.H. | title=Practical Handbook of Curve Design and Generation | publisher=Taylor & Francis | year=1994 | isbn=978-0-8493-8916-0 | url=https://books.google.com/books?id=PVKXqob2dhAC&pg=PA241 | access-date=2022-03-03 | page=241}}</ref> A spiral wound around a helix,<ref name="Wolfram MathWorld 2002">{{cite web |date=2002-09-13 |title=Slinky -- from Wolfram MathWorld |url=https://mathworld.wolfram.com/Slinky.html |access-date=2022-03-03 |website=Wolfram MathWorld}}</ref> also known as ''double-twisted helix'',<ref name="Ugajin Ishimoto Kuroki Hirata 2001 pp. 437–451">{{cite journal | last1=Ugajin | first1=R. | last2=Ishimoto | first2=C. | last3=Kuroki | first3=Y. | last4=Hirata | first4=S. | last5=Watanabe | first5=S. | title=Statistical analysis of a multiply-twisted helix | journal=Physica A: Statistical Mechanics and Its Applications | publisher=Elsevier BV | volume=292 | issue=1–4 | year=2001 | issn=0378-4371 | doi=10.1016/s0378-4371(00)00572-0 | pages=437–451| bibcode=2001PhyA..292..437U }}</ref> represents objects such as [[coiled coil filament]]s.

=== Conical spirals ===
[[File:Spiral-cone-arch-s.svg|thumb|upright=0.8|Conical spiral with Archimedean spiral as floor plan]]
{{main|conical spiral}}
If in the <math>x</math>-<math>y</math>-plane a spiral with parametric representation 
:<math>x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi</math> 
is given, then there can be added a third coordinate <math>z(\varphi)</math>, such that the now space curve lies on the [[cone]] with equation <math>\;m(x^2+y^2)=(z-z_0)^2\ ,\  m>0\;</math>:
* <math>x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\ , \qquad \color{red}{z=z_0 + mr(\varphi)} \ .</math>

Spirals based on this procedure are called '''conical spirals'''.

;Example
Starting with an ''archimedean spiral'' <math>\;r(\varphi)=a\varphi\;</math> one gets the conical spiral (see diagram)
:<math>x=a\varphi\cos\varphi \ ,\qquad y=a\varphi\sin\varphi\ , \qquad z=z_0 + ma\varphi \ ,\quad \varphi \ge 0 \ .</math>

=== 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.