Tractrix
Classic Cartesian curvesThe “drag curve” with a constant-length tangent — rotate it and you get the pseudosphere of hyperbolic geometry.
Adjust parameters
Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
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.
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.
Where you'll see it
The dragged-watch story, pseudosphere models of hyperbolic geometry, tractrix horns in audio, and gear tooth research.