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I Can an envelope curve cut its member curve twice?

  1. Jun 21, 2016 #1
    Can the envelope curve (of a family of curves) intersect a member curve of the family at more than one point?

    It seems possible. Consider the following.

    Screen Shot 2016-06-22 at 4.02.59 am.png
    Each blue line is a member curve and the red line is the envelope curve. If we modify each blue line such that it has a protrusion like a "þ", then it can intersect the red line more than once and still has a point that is tangent to the red line.

    But I have never seen such as example before. So is it not allowed?

    Picture from https://en.wikipedia.org/wiki/Envelope_(mathematics)
  2. jcsd
  3. Jun 21, 2016 #2


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    You would get a different envelope, being tangent to the "þ". You can probably make multiple envelope curves then.
  4. Jun 22, 2016 #3


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    Consider the envelope defined by ##f(x) = 1## for the family of curves ##g_k(x) = sin(x+k)##

    That envelope is tangent to each family member at infinitely many points. My understanding of an "envelope" is that it is not permissible for a family member to extend beyond the envelope. You are only allowed to kiss, not penetrate.
  5. Jun 22, 2016 #4
    This would go well with the English meaning of the word envelope.

    But it is not required in the following definition: an envelope of a family of curves in a plane is a curve that is tangent to each member of the family at some point.
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