Editing Spiral (section)

=== 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>