envelope


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

mx-y-1+m2=0

of the family from the center of the unit circle is

|m0-10-1+m2|m2+(-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-1+m2,  and the equation of each line is given.

y=x+2y=-1y=-x3-2.
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