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enter image description here

I am looking for a formula to make a curve that could mimic loosened thread (from P to Q in the above figure) in 2D. The thread need not be slanted. Wrt x axis I can plot the loosened thread.

Which math formula can model such a curve?

I have tried sine, cosines and harmonics, but it does not seem to be realistic.

Similar question here.

What formula could mimic the following curve?

Vinod
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    Impossible to say unless you're prepared to assert some things about your thread, such as the $y$ values for some set of $x$ values if you were to superimpose it on the $xy$ plane, or the values of its derivatives at some set of points, etc. – H. sapiens rex Mar 20 '25 at 06:49

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