---
tag: 827KR3MVVY
kind: concept
name: Envelope
msc: [53A04, 34A09, 34A26]
sources:
  - url: https://en.wikipedia.org/wiki/Envelope_(mathematics)
same_as:
  wikidata: Q1060372
  wikipedia: Envelope (mathematics)
---

## Summary

The envelope of a one-parameter family of plane curves is a curve that touches every member of the family.
At each of its points it is tangent to one of the curves.
Envelopes give the [[singular solutions|0Q2MDD13G4]] of first-order differential equations, as for [[Clairaut's equation|0T6D4PE51C]], and they appear in optics as caustics.
See the [[definition|HB14PWFCS6]].

## Explanations

### L1 human

Draw many straight lines, each one slightly turned from the previous one.
Although no curve is drawn, the eye sees a curve outlined by the lines: this is the envelope.
For example, the lines at distance $1$ from the origin, in all directions, outline the unit circle.
Each line touches the circle at exactly one point.