# tractrix

Part of the tractrix (locus of A) for the line AB
of length *a*.

The tractrix is a curve, sometimes called the **trajectory curve** or **equitangential
curve**, to which a straight line AB, of length *a*, is always
a tangent at A as B moves along the *x*-axis
from -∞ to +∞. The term is sometimes loosely used to describe
the solid formed by rotation of the curve about the *x*-axis.

The tractrix is the answer to a question asked by the Frenchman Claude Perrault (1613–1688). Perrault is not a giant in the annals of mathematics; in fact, he trained as a doctor and gained a minor reputation as an architect and an anatomist before dying in unusual style as a result of an infection he caught while dissecting a camel. His greatest claim to fame, aside from his connection with the tractrix, is that he was the brother of the author of "Cinderella" and "Puss-in-Boots." In 1676, at about the time Gottfried Leibniz was doing groundbreaking work on the calculus, Perrault placed his pocket-watch on the middle of a table, pulled the end of its chain along the edge of the table, and asked: What is the shape of the curve traced by the watch?

The first known solution was given in a letter to a friend in 1693 by Christiaan Huygens, who also coined the name "tractrix"
from the Latin *tractus* for something that is pulled along. (The corresponding
German name is *hundkurve*, or "hound curve," which makes sense if
you imagine the path a dog might follow on its leash as its master walks
away.) The tractrix can also be found by taking the involute of a catenary. (Imagine a horizontal bar
held at the vertex of the catenary and the point of contact marked as *P*.
When the bar is rolled against the catenary without slipping, the path of *P* is a tractrix.) It is described by the parametric equations: *x* = 1/cosh(*t*), *y* = *t* - tanh(*t*).

The surface of revolution of the tractrix is the pseudosphere, which is the classic model for hyperbolic geometry and one possible three-dimensional analog for the shape of the four-dimensional spacetime in which we live.