# I Can an envelope curve cut its member curve twice?

1. Jun 21, 2016

### Happiness

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.

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. Jun 21, 2016

### Staff: Mentor

You would get a different envelope, being tangent to the "þ". You can probably make multiple envelope curves then.

3. Jun 22, 2016

### jbriggs444

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. Jun 22, 2016

### Happiness

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.