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# 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 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

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

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.

Defines:

envelope

Keywords:

family of curves

Related:

DistanceFromPointToALine

Type of Math Object:

Definition

Major Section:

Reference

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## Mathematics Subject Classification

51N20*no label found*

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