Graph of the Tractrix $y = \ln\dfrac{1+\sqrt{1-x^2}}{x} - \sqrt{1-x^2}$

$y = \ln\dfrac{1 + \sqrt{1-x^2}}{x} - \sqrt{1-x^2}$ is the curve known as the tractrix¹. The name comes from the Latin for "to drag": it is the path traced by an object pulled along by a string of length $1$ whose other end travels down the $y$-axis. Using the inverse hyperbolic cosine it can also be written $y = \operatorname{arcosh}\dfrac{1}{x} - \sqrt{1-x^2}$.

Domain and range

• The domain is $0 < x \leq 1$
• The range is $y \geq 0$
• The function decreases monotonically
• The $y$-axis is a vertical asymptote

At $x = 1$ we have $y = 0$, and as $x \to 0^{+}$ the value $\ln\dfrac{2}{x} - 1$ tends to $+\infty$.

The tangent of constant length

What characterises this curve is a property of its tangent lines.

From the point $(x, y)$ to the $y$-axis the horizontal run is $x$ and the vertical rise along the tangent is $|y'| \cdot x = \sqrt{1-x^2}$, so the tangent segment has constant length.

The fact that the string keeps its length, and always points along the direction of travel, is exactly what this formula encodes.

Monotonicity and concavity

Since $y' < 0$ on $0 < x < 1$ the curve decreases throughout. At $x = 1$ we have $y' = 0$, so it enters $(1, 0)$ tangent to the $x$-axis. The second derivative $y'' = \dfrac{1}{x^2\sqrt{1-x^2}}$ is always positive, making the curve concave up with no inflection point.

Infinite length, finite volume

The arc length from $(1, 0)$ to the point with abscissa $x$ is $\ln\dfrac{1}{x}$, which diverges as $x \to 0$. The curve approaches the $y$-axis arbitrarily closely but never reaches it, and it is infinitely long, yet the surface and solid it sweeps out remain finite.

The pseudosphere

Revolving the curve about its asymptote produces the surface called the pseudosphere². Because $1 + (y')^2 = \dfrac{1}{x^2}$, the line element is $ds = \dfrac{dx}{x}$.

The whole pseudosphere has the same surface area as the sphere of radius $1$, and exactly half its volume.

Negative curvature and non-Euclidean geometry

The pseudosphere has Gaussian curvature equal to $-1$ everywhere. Where the sphere has constant positive curvature, this surface has constant negative curvature, which is what the prefix refers to. In $1868$ Beltrami showed that the parallel postulate fails on it, so that the non-Euclidean geometry of Lobachevsky is realised locally on this surface.

History

Perrault posed the question of what curve a pocket watch describes when dragged across a table by its chain, and Newton and Huygens solved it. The name tractrix is due to Huygens in $1692$.

1. Tractrix, Wikipedia
2. Pseudosphere, Wikipedia