Can an envelope curve cut its member curve twice?

In summary, the conversation discusses the possibility of an envelope curve intersecting a member curve of a family of curves at more than one point. It is suggested that by modifying the member curve to have a protrusion, it can intersect the envelope curve multiple times while still maintaining a point of tangency. However, the concept of an envelope does not necessarily restrict the member curves from extending beyond the envelope. The definition of an envelope allows for tangency at multiple points, but does not explicitly forbid penetration.
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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)
 
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  • #2
You would get a different envelope, being tangent to the "þ". You can probably make multiple envelope curves then.
 
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  • #3
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.
 
  • #4
jbriggs444 said:
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.

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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1. Can an envelope curve cut its member curve twice?

It is possible for an envelope curve to cut its member curve twice, as long as the member curve is self-intersecting or has multiple branches. This typically occurs when the envelope curve is significantly larger than the member curve.

2. How is the envelope curve related to the member curve?

The envelope curve is the locus of the points of tangency between the member curve and a family of curves. This means that the envelope curve is formed by the points where the slopes of the member curve and the family of curves are equal.

3. What is the purpose of studying envelope curves?

Studying envelope curves allows us to understand the relationship between a curve and its family of curves. It can also provide insights into the behavior of a curve and its properties, such as self-intersection and multiple branches.

4. Are there real-world applications for envelope curves?

Yes, envelope curves have various applications in fields such as engineering, mathematics, and computer science. For example, they can be used to model the motion of a pendulum or the shape of a bridge's arch.

5. Is there a specific method for finding the envelope curve of a given member curve?

Yes, there are several methods for finding the envelope curve of a given member curve, including the method of moving tangents, the method of equidistance, and the method of parallel tangents. These methods involve finding the points of tangency between the member curve and a family of curves, and then constructing the envelope curve using these points.

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