Darboux and the Dog's Tail

The master (here the locator) is at the origin, the dog is on the horizontal axis, and the cyan leash joins them. The master decides to head to the north along the 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.

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The architect Claude Perrault (1613–1688) posed the problem of the tractrix and Christian Huygens (1629–1695) solved it in 1692. Later, Leibniz, Johann Bernoulli, and others studied the tractrix. This well-known curve has important applications in engineering.
Rotating the tractrix around its axis generates Beltrami's pseudosphere, a valuable help in visualizing the hyperbolic plane.
Darboux (1842–1917) made significant contributions to geometry and analysis. The transformation, applied here on a line, extends the family of involutes, evolutes, and the like and gives rise to further developments, especially today in discrete geometry.
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