The curve of the obstinate dog (2)

In a previous post I talked about the tractrix: the path followed by a dog if dragged by its lead.

I mentioned that this curve has a fourth highly attractive feature, to be covered another time – the main reason being that to have 2 of these animations together could lead to all sorts of peculiar side effects.

We have our tractrix in red, and we consider each point of this curve in turn. At any such point we draw the ‘normal’: the straight line that is at right angles to the curve. If this is done for all points then a new curve is generated, shown as a bold blue line.

This turns out to be the catenary: the U-shaped curve that a hanging chain or cable naturally assumes when supported only at its ends: pretty cool.

As ever I am grateful for wikipedia for bringing this altogether into one easy-to-remember sentence with which you can impress your friends:

The envelope of the normals of the tractrix – its evolute – is the catenary.

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