# Tractrix

Tim Hoffmann, Jürgen Richter-Gebert

### 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.

### 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