|
(1) |
Using
|
(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
|
(9) |
Then the above equations become
which are equivalent to the form of the original equation if
|
(12) |
|
(13) |
|
(14) |
where only solutions with the minus sign in exist. Solving
gives the values summarized in the following table.
|
|
|
1 |
0.2744106319... |
|
2 |
0.1642700512... |
|
3 |
0.1218322508... |
|
4 |
0.0984064967... |
|
5 |
0.0832810611... |
|
6 |
0.0725974881... |
|
7 |
0.0645958183... |
|
8 |
0.0583494073... |
|
9 |
0.0533203211... |
|
10 |
0.0491732529... |
|
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