Editing Logarithmic spiral (section)

==Properties==
[[File:Spiral-log-st-se.svg|thumb|upright=0.8|Definition of slope angle and sector]]
[[File:Logspiral.gif|thumb|Animation showing the constant angle between an intersecting circle centred at the origin and a logarithmic spiral.]]
The logarithmic spiral <math> r=a e^{k\varphi} \;,\; k\ne 0,</math> has the following properties (see [[Spiral]]):

* '''[[Pitch angle of a spiral|Pitch angle]]''': <math> \tan\alpha=k\quad ( {\color{red}{\text{constant !}}} )</math> {{pb}} with pitch angle <math>\alpha</math> (see diagram and animation).{{pb}}(In case of <math>k=0</math> angle <math> \alpha</math> would be 0 and the curve a circle with radius <math>a</math>.)
* '''Curvature''': <math> \kappa=\frac{1}{r\sqrt{1+k^2}}=\frac{\cos \alpha}{r}</math>
* '''Arc length''': <math> L(\varphi_1,\varphi_2)=\frac{\sqrt{k^2+1}}{k}\big(r(\varphi_2)-r(\varphi_1)\big)= \frac{r(\varphi_2)-r(\varphi_1)}{\sin \alpha}</math>{{pb}}Especially: <math>\ L(-\infty,\varphi_2)=\frac{r(\varphi_2)}{\sin \alpha}\quad  ({\color{red}{\text{finite !}}})\; </math>,  if <math>k > 0</math>. {{pb}} This property was first realized by [[Evangelista Torricelli]] even before [[calculus]] had been invented.<ref>{{cite book | title = The history of the calculus and its conceptual development | author = Carl Benjamin Boyer | publisher = Courier Dover Publications | year = 1949 | isbn = 978-0-486-60509-8 | page = 133 | url = https://books.google.com/books?id=KLQSHUW8FnUC&pg=PA133 }}</ref>
* '''Sector area:''' <math> A=\frac{r(\varphi_2)^2-r(\varphi_1)^2}{4k}</math>
* '''Inversion:''' [[Circle inversion]] (<math>r\to 1/r</math>) maps the logarithmic spiral <math> r=a e^{k\varphi} </math> onto the  logarithmic spiral <math> r=\tfrac{1}{a} e^{-k\varphi} \, .</math>
[[File:Spiral-log-a-1-5.svg|thumb|Examples for <math>a= 1,2,3,4,5</math>]]
* '''Rotating, scaling''': Rotating the spiral by angle <math>\varphi_0</math> yields the spiral <math>r=ae^{-k\varphi_0}e^{k\varphi}</math>, which is the original spiral uniformly scaled (at the origin) by <math>e^{-k\varphi_0}</math>. {{pb}}Scaling by <math>\;e^{kn2\pi}\; , n=\pm 1,\pm2,...,\;</math> gives the ''same'' curve.
* '''[[Self-similarity]]''': A result of the previous property: {{pb}}A scaled logarithmic spiral is [[Congruence (geometry)|congruent]] (by rotation) to the original curve. {{pb}}''Example:'' The diagram shows spirals with slope angle <math>\alpha=20^\circ</math> and <math>a=1,2,3,4,5</math>. Hence they are all scaled copies of the red one. But they can also be generated by rotating the red one by angles <math>-109^\circ,-173^\circ,-218^\circ,-253^\circ</math> resp.. All spirals have no points in common (see property on ''complex exponential function'').
* '''Relation to other curves:''' Logarithmic spirals are congruent to their own [[involute]]s, [[evolute]]s, and the [[pedal curve]]s based on their centers.
* '''Complex exponential function''': The [[exponential function]] exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at <math>0</math>: <math display="block">z(t)=\underbrace{(kt+b)\; +it}_{\text{line}}\quad  \to\quad  e^{z(t)}=e^{kt+b}\cdot e^{it}= \underbrace{e^b e^{kt}(\cos t+i\sin t)}_{\text{log. spiral}} </math> The pitch angle <math>\alpha </math> of the logarithmic spiral is the angle between the line and the imaginary axis.