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  "body": "\n## Summary\n\nThe envelope of a one-parameter family of plane curves is a curve that touches every member of the family.\nAt each of its points it is tangent to one of the curves.\nEnvelopes give the [[singular solutions|0Q2MDD13G4]] of first-order differential equations, as for [[Clairaut's equation|0T6D4PE51C]], and they appear in optics as caustics.\nSee the [[definition|HB14PWFCS6]].\n\n## Explanations\n\n### L1 human\n\nDraw many straight lines, each one slightly turned from the previous one.\nAlthough no curve is drawn, the eye sees a curve outlined by the lines: this is the envelope.\nFor example, the lines at distance $1$ from the origin, in all directions, outline the unit circle.\nEach line touches the circle at exactly one point.\n",
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