Mathematica Notebook for This Page.
above: Tractrix.
Mathematica Notebook for This Page.
Quote from Robert C Yates (1952):
Studied by Huygens↗ in 1692 and later by Leibnitz↗, Johann Bernoulli↗, Liouvlle, and Beltrami↗. Also called Tractory and Equitangential Curve.
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.
Tracing a tractrix.
Parametric:
{Log[Sec[t] + Tan[t]] - Sin[t], Cos[t]},
-π/2 < t < π/2. 
tractrix.gcf.
Cartesian: x == Log[(1 -Sqrt[1^2-y^2])/y] + Sqrt[1^2-y^2].
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.
The evolute of tractrix is catenary, conversely, the involute of catenary is tractrix. The figure on the left connect each point on the tracttrix to its center of tangent circles, thus forming its evolute. On the right, the normals of the tractrix is draw, and their envelope forms its evolute.
This animation shows the trace of a point's center of tangent circle to form a catenary.
Moving Tangent Circle
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.
pseudosphere.nb.zip.
See also: pseudosphere.
The tractrix is a ideal shape for a speaker horn. See Horn speaker↗.
See: Websites on Plane Curves, Printed References On Plane Curves.
The MacTutor History of Mathematics archive↗.
Robert Yates: Curves and Their Properties.
Wikipedia: Tractrix↗.
© 1995-2008 by Xah Lee.
