Darboux and the Dog's Tail

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The master (here the locator) is at the origin, the dog is on the horizontal

x

axis, and the cyan leash joins them. The master decides to head to the north along the

y

axis and has to drag the reluctant dog. The blue curve traced out by the dog is called a tractrix. The other half of the curve follows from symmetry.

However, the dog is not a point! If the dog is long and the leash short, the collar moves on the tractrix, but what curve does the dog's tail follow? This red curve, sometimes called Euler's loop, is obtained from the tractrix by a Darboux transformation.