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

The envelopeMathworldPlanetmath 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-1+m2,  with m the parameter, may be justified geometrically.  It is the open (http://planetmath.org/OpenSet) lower semicircle of the unit circle.  Indeed, the distanceMathworldPlanetmath of any line


of the family from the center of the unit circle is


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-1+m2,  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