(1) |

(2) |

(3) |

and

(4) |

so

(5) |

and the Tangent Vector is given by

(6) |

The coordinates of the Evolute are therefore

(7) | |||

(8) |

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) |

(10) | |||

(11) |

which are equivalent to the form of the original equation if

(12) |

(13) |

(14) |

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

1999-05-25