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Logarithmic Spiral Evolute


\begin{displaymath}
R={(r^2+{r_\theta}^2)^{3/2}\over r^2+2r^2{r_\theta}^2-rr_{\theta\theta}}.
\end{displaymath} (1)

Using
\begin{displaymath}
r=ae^{b\theta} \quad r_\theta=abe^{b\theta}\quad r_{\theta\theta}=ab^2e^{b\theta}
\end{displaymath} (2)

gives
$\displaystyle R$ $\textstyle =$ $\displaystyle {(a^2e^{2b\theta}+a^2b^2e^{2b\theta})^{3/2}\over (ae^{b\theta})^2 +2(abe^{b\theta})^2
-(ab^{b\theta})(ab^2e^{b\theta})}$  
  $\textstyle =$ $\displaystyle {(1+b^2)^{3/2}a^3e^{3b\theta}\over 2a^2b^2e^{2b\theta}+a^2e^{2b\theta}-a^2b^2e^{2b\theta}}$  
  $\textstyle =$ $\displaystyle {(1+b^2)^{3/2}a^3 e^{3b\theta}\over a^2b^2e^{2b\theta}+a^2e^{2b\theta}}
= {(1+b^2)^{3/2}a^3 e^{3b\theta}\over a^2(1+b^2)e^{2b\theta}}$  
  $\textstyle =$ $\displaystyle a\sqrt{1+b^2}\, e^{b\theta}$ (3)

and
$\displaystyle \left[\begin{array}{c}x\\  y\end{array}\right]$ $\textstyle =$ $\displaystyle \left[\begin{array}{c}ae^{b\theta}\cos\theta\\  ae^{b\theta}\sin\theta\end{array}\right]$  
$\displaystyle \left[\begin{array}{c}x'\\  y'\end{array}\right]$ $\textstyle =$ $\displaystyle \left[\begin{array}{c}abe^{b\theta}\cos\theta-ae^{b\theta}\sin\theta\\
abe^{b\theta}\sin\theta+ae^{b\theta}\cos\theta\end{array}\right]\nonumber$  
  $\textstyle =$ $\displaystyle ae^{b\theta}\left[\begin{array}{c}b\cos\theta-\sin\theta\\  b\sin\theta+\cos\theta\end{array}\right],$ (4)

so
$\displaystyle \vert{\bf r}'\vert$ $\textstyle =$ $\displaystyle ae^{b\theta} \sqrt{(b\cos\theta-\sin\theta)^2+(b\sin\theta+\cos\theta)^2}$  
  $\textstyle =$ $\displaystyle ae^{b\theta} \sqrt{1+b^2},$ (5)

and the Tangent Vector is given by
$\displaystyle \hat {\bf T}$ $\textstyle =$ $\displaystyle {{\bf r}'\over \vert{\bf r}'\vert} = {1\over ae^{b\theta} \sqrt{1...
...gin{array}{c}ae^{b\theta}\cos\theta\\  ae^{b\theta}\sin\theta\end{array}\right]$  
  $\textstyle =$ $\displaystyle {1\over\sqrt{1+b^2}}\left[\begin{array}{c}\cos\theta\\  \sin\theta\end{array}\right].$ (6)

The coordinates of the Evolute are therefore
$\displaystyle \xi$ $\textstyle =$ $\displaystyle -abe^{b\theta}\sin\theta$ (7)
$\displaystyle \eta$ $\textstyle =$ $\displaystyle abe^{b\theta}\cos\theta.$ (8)

So the Evolute is another logarithmic spiral with $a'\equiv ab$, as first shown by Johann Bernoulli. However, in some cases, the Evolute is identical to the original, as can be demonstrated by making the substitution to the new variable
\begin{displaymath}
\theta\equiv \phi-{\textstyle{1\over 2}}\pi\pm 2n\pi.
\end{displaymath} (9)

Then the above equations become
$\displaystyle \xi$ $\textstyle =$ $\displaystyle -abe^{b(\phi-\pi/2\pm 2n\pi)}\sin(\phi-\pi/2\pm 2n\pi)$  
  $\textstyle =$ $\displaystyle abe^{b\phi}e^{b(-\pi/2\pm 2n\pi)}\cos\phi$ (10)
$\displaystyle \eta$ $\textstyle =$ $\displaystyle abe^{b(\phi-\pi/2\pm 2n\pi)}\cos(\phi-\pi/2\pm 2n\pi)$  
  $\textstyle =$ $\displaystyle abe^{b\phi}e^{b(-\pi/2\pm 2n\pi)}\sin\phi,$ (11)

which are equivalent to the form of the original equation if
\begin{displaymath}
be^{b(-{\textstyle{1\over 2}}\pi\pm 2n\pi)}=1
\end{displaymath} (12)


\begin{displaymath}
\ln b+b(-{\textstyle{1\over 2}}\pi\pm 2n\pi)=0
\end{displaymath} (13)


\begin{displaymath}
{\ln b\over b} = {\textstyle{1\over 2}}\pi\mp 2n\pi = -(2n-{\textstyle{1\over 2}})\pi,
\end{displaymath} (14)

where only solutions with the minus sign in $\mp$ exist. Solving gives the values summarized in the following table.

$n$ $b_n$ $\psi=\cot^{-1} b_n$
1 0.2744106319... $74^\circ 39' 18.53''$
2 0.1642700512... $80^\circ 40' 16.80''$
3 0.1218322508... $83^\circ 03' 13.53''$
4 0.0984064967... $84^\circ 22' 47.53''$
5 0.0832810611... $85^\circ 14' 21.60''$
6 0.0725974881... $85^\circ 50' 51.92''$
7 0.0645958183... $86^\circ 18' 14.64''$
8 0.0583494073... $86^\circ 39' 38.20''$
9 0.0533203211... $86^\circ 56' 52.30''$
10 0.0491732529... $87^\circ 11' 05.45''$


References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 60-64, 1991.



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© 1996-9 Eric W. Weisstein
1999-05-25