- #1

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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)