Editing Spiral (section)

=== Examples ===

Some of the most important sorts of two-dimensional spirals include:

* The [[Archimedean spiral]]: <math>r=a \varphi </math>
* The [[hyperbolic spiral]]: <math>r = a/ \varphi</math>
* [[Fermat's spiral]]: <math>r= a\varphi^{1/2}</math>
* The [[Lituus (mathematics)|lituus]]: <math>r = a\varphi^{-1/2}</math>
* The [[logarithmic spiral]]: <math>r=ae^{k\varphi}</math> 
* The [[Cornu spiral]] or ''clothoid''
* The [[Fibonacci spiral]] and [[golden spiral]]
* The [[Spiral of Theodorus]]: an approximation of the Archimedean spiral composed of contiguous right triangles
* The [[involute]] of a circle
<gallery>
Image:Archimedean spiral.svg|Archimedean spiral
Image:Hyperspiral.svg|hyperbolic spiral
Image:Fermat's spiral.svg|Fermat's spiral
Image:Lituus.svg|lituus
Image:Logarithmic Spiral Pylab.svg|logarithmic spiral
Image:Cornu Spiral.svg|Cornu spiral
Image:Spiral of Theodorus.svg|spiral of Theodorus
Image:Fibonacci_spiral.svg|Fibonacci Spiral (golden spiral)
Image:Archimedean-involute-circle-spirals-comparison.svg|The involute of a circle (black) is not identical to the Archimedean spiral (red).
</gallery>
[[File:Schraublinie-hyp-spirale.svg|thumb|upright=0.6|Hyperbolic spiral as central projection of a helix]]

An ''Archimedean spiral'' is, for example, generated while coiling a carpet.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Archimedean Spiral|url=https://mathworld.wolfram.com/ArchimedeanSpiral.html|access-date=2020-10-08|website=mathworld.wolfram.com|language=en}}</ref>

A ''hyperbolic spiral'' appears as image of a helix with a special central projection (see diagram). A hyperbolic spiral is some times called ''reciproke'' spiral, because it is the image of an Archimedean spiral with a circle-inversion (see below).<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Hyperbolic Spiral|url=https://mathworld.wolfram.com/HyperbolicSpiral.html|access-date=2020-10-08|website=mathworld.wolfram.com|language=en}}</ref>

The name ''logarithmic spiral'' is due to the equation <math>\varphi=\tfrac{1}{k}\cdot \ln \tfrac{r}{a}</math>. Approximations of this are found in nature.

Spirals which do not fit into this scheme of the first 5 examples:

A ''Cornu spiral'' has two asymptotic points.<br>
The ''spiral of Theodorus'' is a polygon.<br>
The ''Fibonacci Spiral'' consists of a sequence of circle arcs.<br>
The ''involute of a circle'' looks like an Archimedean, but is not: see [[Involute#Examples]].