What is the equation of the Tractrix?

Parametric equations: x = a(t - tanh t), y = a/(cosh t). a is the length of the leash; the x-axis is the line it is pulled along.

What is the area of the Tractrix?

Area (between both branches and the asymptote): A = (π a^2)/2. a is the length of the leash; the x-axis is the line it is pulled along.

Classic Cartesian curves

Tractrix

The “drag curve” with a constant-length tangent — rotate it and you get the pseudosphere of hyperbolic geometry.

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a (leash length)1

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Formulas and key values

Parametric equations: $x = a(t - \tanh t)$ $y = \frac{a}{\cosh t}$
Asymptote: $y = 0$
Tangent segment to the asymptote: $a$
Area · between both branches and the asymptote: $A = \frac{\pi a^2}{2}$

a is the length of the leash; the x-axis is the line it is pulled along.

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01 · Definition

What is the Tractrix?

Drag a reluctant object by a leash of length a while you walk along a straight line: the object traces the tractrix. Claude Perrault posed the puzzle in Paris around 1670 with his pocket watch on a table; Huygens named the curve and Leibniz analyzed it in 1693. Its defining property is built into the story: the tangent segment from the curve to the pulling line always has length exactly a.

The curve’s afterlife is remarkable. Revolving a tractrix about its asymptote produces the pseudosphere, a trumpet-shaped surface of constant negative curvature on which Beltrami realized hyperbolic geometry concretely in 1868 — turning “imaginary geometry” into something you can hold.

02 · Properties

Key properties

The tangent segment from any point to the asymptote has constant length a.
The x-axis is an asymptote; the curve starts at the cusp (0, a).
Its evolute (envelope of normals) is the catenary.
Rotating about the asymptote gives the pseudosphere: constant Gaussian curvature −1/a².
The pseudosphere has finite area 4πa² and finite volume despite its infinite length.

03 · Questions

Tractrix: common questions

What is the equation of the Tractrix?: Parametric equations: x = a(t - tanh t), y = a/(cosh t). a is the length of the leash; the x-axis is the line it is pulled along.
What is the area of the Tractrix?: Area (between both branches and the asymptote): A = (π a^{^2})/2. a is the length of the leash; the x-axis is the line it is pulled along.

04 · Keep exploring

All curves · 60

Classic Cartesian curves

Catenary

Polar curves

Cissoid of Diocles

Spirals