envelope

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An envelope of a set of curves is a curve that is tangent to these curves.

Examples are the following:

  • an astroid is the envelope of straight lines
  • an astroid is the envelope of co-axial ellipses for which the sum of the major and minor axes is constant
  • a cardioid is the envelope of the chords of a circle
  • a cardioid is the envelope of circles
  • a catacaustic is the envelope of reflected rays
  • a deltoid is the envelope of the so-called Simson lines
  • a diacaustic is the envelope of refracted rays
  • the lemniscate is the envelope of circles with their centers on a rectangular hyperbola
  • the nephroid is the envelope of a set of circles with centers on a base circle and tangent to a diameter (of the base circle)

In fact, the envelope of the normal of a curve is the same as the evolute of the curve.