![\begin{displaymath}
R={(r^2+{r_\theta}^2)^{3/2}\over r^2+2r^2{r_\theta}^2-rr_{\theta\theta}}.
\end{displaymath}](l2_656.gif) |
(1) |
Using
![\begin{displaymath}
r=ae^{b\theta} \quad r_\theta=abe^{b\theta}\quad r_{\theta\theta}=ab^2e^{b\theta}
\end{displaymath}](l2_657.gif) |
(2) |
gives
and
so
and the Tangent Vector is given by
The coordinates of the Evolute are therefore
So the Evolute is another logarithmic spiral with
, 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}](l2_679.gif) |
(9) |
Then the above equations become
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}](l2_684.gif) |
(12) |
![\begin{displaymath}
\ln b+b(-{\textstyle{1\over 2}}\pi\pm 2n\pi)=0
\end{displaymath}](l2_685.gif) |
(13) |
![\begin{displaymath}
{\ln b\over b} = {\textstyle{1\over 2}}\pi\mp 2n\pi = -(2n-{\textstyle{1\over 2}})\pi,
\end{displaymath}](l2_686.gif) |
(14) |
where only solutions with the minus sign in
exist. Solving
gives the values summarized in the following table.
![$n$](l2_102.gif) |
![$b_n$](l2_688.gif) |
![$\psi=\cot^{-1} b_n$](l2_689.gif) |
1 |
0.2744106319... |
![$74^\circ 39' 18.53''$](l2_690.gif) |
2 |
0.1642700512... |
![$80^\circ 40' 16.80''$](l2_691.gif) |
3 |
0.1218322508... |
![$83^\circ 03' 13.53''$](l2_692.gif) |
4 |
0.0984064967... |
![$84^\circ 22' 47.53''$](l2_693.gif) |
5 |
0.0832810611... |
![$85^\circ 14' 21.60''$](l2_694.gif) |
6 |
0.0725974881... |
![$85^\circ 50' 51.92''$](l2_695.gif) |
7 |
0.0645958183... |
![$86^\circ 18' 14.64''$](l2_696.gif) |
8 |
0.0583494073... |
![$86^\circ 39' 38.20''$](l2_697.gif) |
9 |
0.0533203211... |
![$86^\circ 56' 52.30''$](l2_698.gif) |
10 |
0.0491732529... |
![$87^\circ 11' 05.45''$](l2_699.gif) |
References
Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press,
pp. 60-64, 1991.
© 1996-9 Eric W. Weisstein
1999-05-25