Tractrix

tractrix
Tractrix.

Mathematica Notebook for This Page.

History

From Robert Yates:

Studied by Huygens in 1692 and later by Leibnitz, Johann Bernoulli, Liouvlle, and Beltrami. Also called Tractory and Equitangential Curve.

Description

Tracktrix (equitangential curve, tractory) is a curve such that any tangent segment from the tangent point on the curve to the curve's asymptote have constant length.

Suppose a bicycle with front wheel at the {0,0} and back wheel at {0,1}. The front wheel is turned to head East. The track traced by the back wheel is the tractrix. Note that the tractrix's asymptote cuts its tangents into segments of equal length.

tractrix tractrixGen
Tractrix Tracing

Formula

Parametric: {Log[Sec[t] + Tan[t]] - Sin[t], Cos[t]}, -π/2 < t < π/2.

Cartesian: x == Log[(1 -Sqrt[1^2-y^2])/y] + Sqrt[1^2-y^2].

Properties

Orthogonal to Circles

The tractrix is orthogonal to a set of circles centered on the tractrix's asymptote, all having radius radius R. R is the same as the tractrix's constant length tangent segment.

tractrix orthogonal to circles
Tractrix as a curve orthogonal to a particular set of circles.

Tractrix and Catenary

The evolute of tractrix is catenary, conversely, the involute of catenary is tractrix.

tractrix tractrix
Left: The normals of the tractrix. Right: The normals of a tractrix stopping at the center of its osculating circle.
tractrix tractrixCatOsculating
The trace of a point's center of tangent circle to form a catenary

Pseudosphere

The surface of revolution of tractrix around its asymptote is called pseudosphere. Eugenio Beltrami in 1868 showed that pseudosphere provided a model for hyperbolic geometry. It is a surface of constant negative Gaussian curvature.

tractrix
The pseudosphere. pseudosphere.nb.zip.

See also: pseudosphere.

Applications

The tractrix is a ideal shape for a speaker horn. See Horn speaker.

Related Web Sites

See: Websites on Plane Curves, Printed References On Plane Curves.

The MacTutor History of Mathematics archive

Robert Yates: Curves and Their Properties.

Tractrix.