Tractrix and truck tricks

by Burkard Polster and Marty Ross

The Age, 6 May 2013

As we have written of time and time again, beautiful mathematics is everywhere. And, last Thursday, opening an innocent kitchen cupboard, there it was: some very clever tea bags had formed themselves into a famous mathematical curve, the tractrix.

Indeed, tractrices had been stalking your Maths Masters all week. Earlier, while practising some card flourishes and flipping over a ribbon spread, there was another:

Then, while toppling domino chains with some junior Maths Masters, another tractrix appeared:

Finally, your Masters Masters found themselves caught behind a monster truck. In order to drag its superlong trailer through a T-junction the truck cabin had to begin traversing the corner at right angles to the trailer:

The rear (red) wheel passed through a puddle and traced out a tractrix on the dry street ahead.

The above curves are all striking and they appear to be basically the same. But what exactly makes a tractrix a tractrix?

The curve was first considered by 17th century French anatomist and architect Claude Perrault (the much less famous brother of Charles Perrault, author of Little Red Riding Hood and other classic fairy tales). He placed his watch in the middle of a table and pulled the end of the watch chain along the edge of the table. Perrault then asked for the shape of the path traced out by the watch. This is identical to our truck scenario above.

Perrault was not a mathematician and it took the great Sir Isaac Newton and Gottfried Leibniz, and later Christiaan Huygens, to solve the problem. The name "tractrix" for the curve is also due to Leibniz. It is derived from the Latin trahere, meaning to pull, and which similarly gives us the word "tractor".

The key to determining the tractrix is to realise that the rear red wheel (or watch) is a fixed distance behind the green front wheel (chain end), and that the truck trailer (watch chain) always indicates the direction of motion of the rear red wheel. That is, we know the tangent to the tractrix at any point and our problem then is to determine the tractrix itself.

Naturally enough, 17th century mathematicians referred to such a question as "a problem of inverse tangents". These days we would express the problem as a differential equation. We shan't give the details here, but it is not difficult to determine and to solve the tractrix differential equation (and the exercise would make the basis of an excellent Mathematical Methods SAC).

Once you have the notion in mind, you'll begin spotting tractrices everywhere. Moreover, the tractrix path and its generalizations known as tractional motion, where the front wheel needn't travel in a straight line, have become an indispensible tool in the design of street intrersections, parking garages and the like.

Suppose that you want to check whether your newly designed roundabout will accommodate a certain size truck. One fun method is to build the roundabout, drive a truck through it, survey the damage and adjust the roundabout accordingly.

Less fun but a lot cheaper is to perform a swept path analysis. As indicated in the above diagram (created with the software package Autotrack), this consists of driving a virtual version of the truck through the roundabout and calculating the regions swept out by the different components of the truck. These swept out regions are typically bordered by tractional curves.

There is much more to the tractrix, and we hope to write soon about an astonishing 2D world that the tractrix generates. But for this week we'll stick to the trucks, and we'll close with a simple safety message. Doesn’t the tractrix demonstrate quite dramatically why you should heed those "do not overtake" signs displayed on big trucks?

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.