Let two Circles of Radii and and centered at and intersect
in a Lens-shaped region. The equations of the two circles are

(1) | |||

(2) |

Combining (1) and (2) gives

(3) |

(4) |

(5) |

(6) |

giving a length of

(7) |

This same formulation applies directly to the Sphere-Sphere Intersection problem.

To find the Area of the asymmetric ``Lens'' in which the Circles intersect,
simply use the formula for the circular Segment of radius and triangular height

(8) |

(9) | |||

(10) |

The result is

(11) |

The limiting cases of this expression can be checked to give 0 when and

(12) | |||

(13) |

when , as expected. In order for half the area of two Unit Disks () to overlap, set in the above equation

(14) |

© 1996-9

1999-05-26