Logarithmic Spiral Evolute

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

Using

$$r=ae^{b\theta} \quad r_\theta=abe^{b\theta}\quad r_{\theta\theta}=ab^2e^{b\theta}$$ (2)

gives

$$R = {(a^2e^{2b\theta}+a^2b^2e^{2b\theta})^{3/2}\over (ae^{b\theta})^2 +2(abe^{b\theta})^2 -(ab^{b\theta})(ab^2e^{b\theta})}$$

$$= {(1+b^2)^{3/2}a^3e^{3b\theta}\over 2a^2b^2e^{2b\theta}+a^2e^{2b\theta}-a^2b^2e^{2b\theta}}$$

$$= {(1+b^2)^{3/2}a^3 e^{3b\theta}\over a^2b^2e^{2b\theta}+a^2e^{2b\theta}} = {(1+b^2)^{3/2}a^3 e^{3b\theta}\over a^2(1+b^2)e^{2b\theta}}$$

$$= a\sqrt{1+b^2}\, e^{b\theta}$$ (3)

and

$$\left[\begin{array}{c}x\\ y\end{array}\right] = \left[\begin{array}{c}ae^{b\theta}\cos\theta\\ ae^{b\theta}\sin\theta\end{array}\right]$$

$$\left[\begin{array}{c}x'\\ y'\end{array}\right] = \left[\begin{array}{c}abe^{b\theta}\cos\theta-ae^{b\theta}\sin\theta\\ abe^{b\theta}\sin\theta+ae^{b\theta}\cos\theta\end{array}\right]\nonumber$$

$$= ae^{b\theta}\left[\begin{array}{c}b\cos\theta-\sin\theta\\ b\sin\theta+\cos\theta\end{array}\right],$$ (4)

so

$$\vert{\bf r}'\vert = ae^{b\theta} \sqrt{(b\cos\theta-\sin\theta)^2+(b\sin\theta+\cos\theta)^2}$$

$$= ae^{b\theta} \sqrt{1+b^2},$$ (5)

and the Tangent Vector is given by

$$\hat {\bf T} = {{\bf r}'\over \vert{\bf r}'\vert} = {1\over ae^{b\theta} \sqrt{1... ...gin{array}{c}ae^{b\theta}\cos\theta\\ ae^{b\theta}\sin\theta\end{array}\right]$$

$$= {1\over\sqrt{1+b^2}}\left[\begin{array}{c}\cos\theta\\ \sin\theta\end{array}\right].$$ (6)

The coordinates of the Evolute are therefore

$$\xi = -abe^{b\theta}\sin\theta$$ (7)

$$\eta = abe^{b\theta}\cos\theta.$$ (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

$$\theta\equiv \phi-{\textstyle{1\over 2}}\pi\pm 2n\pi.$$ (9)

Then the above equations become

$$\xi = -abe^{b(\phi-\pi/2\pm 2n\pi)}\sin(\phi-\pi/2\pm 2n\pi)$$

$$= abe^{b\phi}e^{b(-\pi/2\pm 2n\pi)}\cos\phi$$ (10)

$$\eta = abe^{b(\phi-\pi/2\pm 2n\pi)}\cos(\phi-\pi/2\pm 2n\pi)$$

$$= abe^{b\phi}e^{b(-\pi/2\pm 2n\pi)}\sin\phi,$$ (11)

which are equivalent to the form of the original equation if

$$be^{b(-{\textstyle{1\over 2}}\pi\pm 2n\pi)}=1$$ (12)

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

$${\ln b\over b} = {\textstyle{1\over 2}}\pi\mp 2n\pi = -(2n-{\textstyle{1\over 2}})\pi,$$ (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.