# 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 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.

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Contributed by: Claude Fabre (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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.

## Permanent Citation

"Darboux and the Dog's Tail"

http://demonstrations.wolfram.com/DarbouxAndTheDogsTail/

Wolfram Demonstrations Project

Published: March 7 2011