Tractrix

Tim Hoffmann, Jürgen Richter-Gebert

Media


Description


The applet shows a discrete version of the classical tractrix construction. In his 1693 Leipziger Acta eruditorum Problem Leibniz asked the question: ''In the xy-plane drag a point $P$ with a tightly strained string $PZ$ of length $a$. The ``drag point'' $Z$ shall propagate along the pasitive $y$-axis, and at the beginning $P$ shall be in $(a,0)$. Which curve is described by $P$?'' For comprehension Leibniz imagined a pocket watch on a chain. As source of the problem he mentioned the Paris Architect Claude Perault. This question can be asked for a generic smooth curve $\gamma$ resulting in a differential equation for the tractrix curve $\hat\gamma$ given by the conditions $\hat\gamma^\prime\parallel\hat\gamma-\gamma = v$ and $\vert v \vert=const$. The tractrix is closely related to the so-called Darboux transform $\tilde \gamma$ of the curve $\gamma$ which can be described as the curve at twice the distance of the tractrix: $\tilde\gamma = \gamma+ 2(\hat\gamma - \gamma) = \gamma + 2v$. It turns out that $\gamma$ and $\tilde \gamma$ are not only in constant distance $2v$ but also arc-length related. This observation is key to the discretization.

Given a polygon asking for a new polygon in constant distance with the same edge lengths gives (after choosing an initial point) for every edge two choices: one makes the edge and its transform a parallelogram the other makes them a parallelogram folded along the diagonal. The second choice gives rise to a dicretization of the Darboux transformation and the polygon half way between the two is called a discrete tractrix. This is what the applet shows. You can drag the (red) points of the original curve as well as the larger yellow one for the initial condition.

See also: [1]

References


  • Tim Hoffmann.
    Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow.
    Discrete Differential Geometry, 2008(38):95–115, 2008.

Prof. Dr. Tim Hoffmann   +

Projects: A02
University: TU München
E-Mail: hoffmant[at]ma.tum.de
Website: http://www-m10.ma.tum.de/bin/view/Lehrstuhl/TimHoffmann


Prof. Dr. Dr. Jürgen Richter-Gebert   +

Projects: CaP, Z02
University: TU München, Zentrum Mathematik, M10, 02.06.054
Address: Boltzmannstr. 3, D-85748 Garching bei München
Tel: +49 (89) 289 18354
E-Mail: richter[at]ma.tum.de
Website: http://www-m10.ma.tum.de/bin/view/Lehrstuhl/RichterGebert