Editing Spiral (section)

==Two-dimensional==
{{main|List of spirals}}

[[File:Six types of spirals.png|thumb|Spirals generated by 6 mathematical relationships between radius and angle.]]

A [[two-dimensional]], or plane, spiral may be easily described using [[polar coordinates]], where the [[radius]] <math>r</math> is a [[monotonic]] [[continuous function]] of angle <math>\varphi</math>:
* <math>r=r(\varphi)\; .</math>
The circle would be regarded as a [[degenerate (mathematics)|degenerate]] case (the [[Function (mathematics)|function]] not being strictly monotonic, but rather [[Constant (mathematics)|constant]]).

In ''<math>x</math>-<math>y</math>-coordinates'' the curve has the parametric representation:
* <math>x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\; .</math>

=== 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]].

=== Geometric properties ===
The following considerations are dealing with spirals, which can be described by a polar equation <math>r=r(\varphi)</math>, especially for the cases <math>r(\varphi)=a\varphi^n</math> (Archimedean, hyperbolic, Fermat's, lituus spirals) and the logarithmic spiral <math>r=ae^{k\varphi}</math>.

[[File:Sektor-steigung-pk-def.svg|thumb|Definition of sector (light blue) and polar slope angle <math>\alpha</math>]]

;Polar slope angle
The angle <math>\alpha</math> between the spiral tangent and the corresponding polar circle (see diagram) is called ''angle of the polar slope'' and <math>\tan \alpha</math> the ''polar slope''.

From [[polar coordinate system#Vector calculus|vector calculus in polar coordinates]] one gets the formula
:<math>\tan\alpha=\frac{r'}{r}\ .</math>

Hence the slope of the spiral <math>\;r=a\varphi^n \;</math>  is
* <math>\tan\alpha=\frac{n}{\varphi}\ .</math>

In case of an ''Archimedean spiral'' (<math>n=1</math>) the polar slope is <math>\; \tan\alpha=\tfrac{1}{\varphi}\ .</math>

In a ''logarithmic spiral'', <math>\ \tan\alpha=k\ </math> is constant.

;Curvature
The curvature <math>\kappa</math> of a curve with polar equation <math>r=r(\varphi)</math> is

:<math>\kappa = \frac{r^2 + 2(r')^2 - r\; r''}{(r^2+(r')^2)^{3/2}}\ .</math>

For a spiral with <math>r=a\varphi^n</math> one gets
* <math>\kappa = \dotsb = \frac{1}{a\varphi^{n-1}}\frac{\varphi^2+n^2+n}{(\varphi^2+n^2)^{3/2}}\ .</math>

In case of <math>n=1</math> ''(Archimedean spiral)''
<math>\kappa=\tfrac{\varphi^2+2}{a(\varphi^2+1)^{3/2}}</math>.<br> 
Only for <math>-1<n<0 </math> the spiral has an ''inflection point''.

The curvature of a ''logarithmic spiral'' <math>\; r=a e^{k\varphi} \;</math> is <math>\; \kappa=\tfrac{1}{r\sqrt{1+k^2}} \; .</math>

;Sector area
The area of a sector of a curve (see diagram) with polar equation <math>r=r(\varphi)</math> is
:<math>A=\frac{1}{2}\int_{\varphi_1}^{\varphi_2} r(\varphi)^2\; d\varphi\ .</math>

For a spiral with equation <math>r=a\varphi^n\; </math> one gets

* <math>A=\frac{1}{2}\int_{\varphi_1}^{\varphi_2} a^2\varphi^{2n}\; d\varphi
=\frac{a^2}{2(2n+1)}\big(\varphi_2^{2n+1}- \varphi_1^{2n+1}\big)\ , \quad \text{if}\quad n\ne-\frac{1}{2},
</math>
:<math>A=\frac{1}{2}\int_{\varphi_1}^{\varphi_2} \frac{a^2}{\varphi}\; d\varphi
=\frac{a^2}{2}(\ln\varphi_2-\ln\varphi_1)\ ,\quad \text{if} \quad n=-\frac{1}{2}\ .</math>

The formula for a ''logarithmic spiral'' <math>\; r=a e^{k\varphi} \;</math> is
<math>\ A=\tfrac{r(\varphi_2)^2-r(\varphi_1)^2)}{4k}\ .</math>

;Arc length
The length of an arc of a curve with polar equation <math>r=r(\varphi)</math> is
:<math>L=\int\limits_{\varphi_1}^{\varphi_2}\sqrt{\left(r^\prime(\varphi)\right)^2+r^2(\varphi)}\,\mathrm{d}\varphi \ .</math>

For the spiral <math>r=a\varphi^n\; </math> the length is

* <math>L=\int_{\varphi_1}^{\varphi_2} \sqrt{\frac{n^2r^2}{\varphi^2} +r^2}\; d\varphi
= a\int\limits_{\varphi_1}^{\varphi_2}\varphi^{n-1}\sqrt{n^2+\varphi^2}d\varphi
\ .</math>
Not all these integrals can be solved by a suitable table. In case of a Fermat's spiral, the integral can be expressed by [[elliptic integral]]s only.

The arc length of a ''logarithmic spiral'' <math>\; r=a e^{k\varphi} \;</math> is <math>\ L=\tfrac{\sqrt{k^2+1}}{k}\big(r(\varphi_2)-r(\varphi_1)\big) \ .</math>

;Circle inversion
The [[Circle inversion|inversion at the unit circle]] has in polar coordinates the simple description: <math>\ (r,\varphi) \mapsto (\tfrac{1}{r},\varphi)\ </math>.

* The image of a spiral <math>\ r= a\varphi^n\ </math> under the inversion at the unit circle is the spiral with polar equation <math>\ r= \tfrac{1}{a}\varphi^{-n}\ </math>. For example: The inverse of an Archimedean spiral is a hyperbolic spiral.
:A logarithmic spiral <math>\; r=a e^{k\varphi} \;</math> is mapped onto the logarithmic spiral <math>\; r=\tfrac{1}{a} e^{-k\varphi} \; .</math>

=== Bounded spirals ===
[[File:Spiral-arctan-1-2.svg|thumb|upright=1.4|Bounded spirals:<br>
<math>r=a \arctan(k\varphi)</math> (left), <br>
<math>r=a (\arctan(k\varphi)+\pi/2) </math> (right)]]
The function <math>r(\varphi)</math> of a spiral is usually strictly monotonic, continuous
and un[[Bounded function|bounded]]. For the standard spirals  <math>r(\varphi)</math> is either a power function or an exponential function. If one chooses for <math>r(\varphi)</math> a ''bounded'' function, the spiral is bounded, too. A suitable bounded function is the [[arctan]] function:

;Example 1
Setting <math>\;r=a \arctan(k\varphi)\;</math> and the choice <math>\;k=0.1, a=4, \;\varphi\ge 0\;</math> gives a spiral, that starts at the origin (like an Archimedean spiral) and approaches the circle with radius <math>\;r=a\pi/2\;</math> (diagram, left).
;Example 2
For <math>\;r=a (\arctan(k\varphi)+\pi/2)\;</math> and <math>\;k=0.2, a=2,\; -\infty<\varphi<\infty\;</math> one gets a spiral, that approaches the origin (like a hyperbolic spiral) and approaches the circle with radius <math>\;r=a\pi\;</math> (diagram, right).