# envelope

Two plane curves^{} are said to touch each other or have a tangency at a point if they have a common tangent line at that point.

The envelope^{} of a family of plane curves is a curve which touches in each of its points one of the curves of the family.

For example, the envelope of the family $y=mx-\sqrt{1+{m}^{2}}$, with $m$ the parameter, may be justified geometrically. It is the open (http://planetmath.org/OpenSet) lower semicircle of the unit circle. Indeed, the distance^{} of any line

$$mx-y-\sqrt{1+{m}^{2}}=0$$ |

of the family from the center of the unit circle is

$$\frac{|m\cdot 0-1\cdot 0-\sqrt{1+{m}^{2}}|}{\sqrt{{m}^{2}+{(-1)}^{2}}}=1,$$ |

whence the line is the tangent to the circle.

Below, the red curve is the lower semicircle of the unit circle, the black lines belong to the family $y=mx-\sqrt{1+{m}^{2}}$, and the equation of each line is given.

Title | envelope |
---|---|

Canonical name | Envelope |

Date of creation | 2013-03-22 17:10:19 |

Last modified on | 2013-03-22 17:10:19 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 23 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 51N20 |

Related topic | DistanceFromPointToALine |

Defines | envelope |