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I'm writing a description that involves two curves behaving (approximately) as shown below. There aren't actually two intersections: they are mutually tangential. Also, I have many such curves, this is just a characteristic example. Something like $x^2$ and $2x^2$ at the origin is another example.

How does one describe the relation between these two curves?

There are two important properties here. The first is that the curves are tangential. i.e. they touch with equal derivatives. The second important feature is that they "move away from each other" on either side of the point where they "touch". So, e.g. if you took the diffence between the two curves (say up is the $y$-axis) then you'd find it is growing.

I had been using the clunky "Curve A is tangential to and curved away from Curve B". Some suggested "Curve A osculates Curve B", but having looked up more about the word osculate, I'm not sure this is right. Although the examples I've seen of osculation would all satisfy my condition, the curve in the picture is not osculation, as I understand...

I did try to dig around for an answer here before asking, but without luck. Apologies if this question is a duplicate. Also, owing to its technical nature, I presume it's better off here than an English Usage SE.

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1 Answer 1

You can read about different orders of contact in this Wikipedia article. For two curves to be osculating, the curvatures at the point of contact are required to coincide. You don't specify the order of "moving away from each other", so your curves could generally be osculating or not. However, in the two examples you give, the image and the case of $x^2$ and $2x^2$, the curvatures of the curves differ, so you could describe these cases by saying that the curves are tangent but not osculating.

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Put another way, "osculating" is usually given the meaning "both curves agree up to the second derivative". –  J. M. Dec 5 '11 at 10:26
    
I guess the case I need to exclude is that the curves are tangent and then cross. So, like $x^3$ vs $0$ at the origin, which is also tangent and not osculating... –  Warrick Dec 5 '11 at 10:47
    
Actually, I'd say that the horizontal axis and $x^3$ are osculating... –  J. M. Dec 5 '11 at 11:10
    
Ah, you're absoultely right. –  Warrick Dec 5 '11 at 11:16
    
I don't think "tangent but not osculating" would work in all cases. Consider, e.g., $y=x^4$ and $y=2x^4$. I think "curves away from each other" or "the distance between the curves increases" would be the best way to describe this. –  David Mitra Dec 5 '11 at 11:30

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