Editing Envelope (mathematics) (section)

{{short description|Curve external to a family of curves in geometry}}
{{About||the envelope of an oscillating signal|Envelope (waves)|the abstract concept|Envelope (category theory)}}
[[Image:EnvelopeAnim.gif|upright=1.5|thumb|Construction of the envelope of a family of curves.]]

In [[geometry]], an '''envelope''' of a planar [[family of curves]] is a [[curve]] that is [[tangent]] to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two "[[infinitesimal]]ly adjacent" curves, meaning the [[Limit (mathematics)|limit]] of intersections of nearby curves. This idea can be [[Universal generalization|generalized]] to an envelope of [[Surface (mathematics)|surfaces]] in space, and so on to higher dimensions.

To have an envelope, it is necessary that the individual members of the family of curves are [[Differentiable manifold|differentiable curve]]s as the concept of tangency does not apply otherwise, and there has to be a [[Smoothness|smooth]] transition proceeding through the members. But these conditions are not sufficient – a given family may fail to have an envelope. A simple example of this is given by a family of concentric circles of expanding radius.