Magazine article Mathematics Teaching

Sand Curves and Symbolic Geometry

Magazine article Mathematics Teaching

Sand Curves and Symbolic Geometry

Article excerpt

Philip Todd starts with the idea ... Everybody knows that, if you tie a string onto two sticks and draw keeping the string taut, you get an ellipse

An Oregon beach at low tide provides an ideal sandbox. If you find an old piece of netting and pick up some driftwood, you can't resist using it for geometry.

This combination of circumstances occurred to me last autumn, when I was looking for good examples to demonstrate the use of the new symbolic geometry program Geometry Expressions [1]. Unlike other geometry software, Geometry Expressions derives the algebraic equations implied by your geometry. I wanted to draw some ellipses, photograph them, modei them in the software, and examine their equations.

I started by attaching my string to two sticks on a beach and drawing round with a third stick keeping the string tight.

Half an Ellipse

Of course, at first I was indeed tracing an ellipse.

This can be modelled in Geometry Expressions directly. Let's assume that the string is length L and that the sticks are distance 2a apart. We can set the locations of the sticks to be the points (-a, 0) and (a, 0). We then draw a triangle representing one position of the string. We make the distance to the apex from one of the sticks t, twill vary and be a parameter of the motion. We make the distance to the other stick L -1. We then ask the software to create the locus of the apex point as t varies, see Figure 2.

The unique aspect of Geometry Expressions is that it can now give the equation of the curve in terms of X, Y, and L. By examining the equation shown in Figure 2, can you determine the width and height of the ellipse? Can you verify these equations by thinking about the string?

Our software has only drawn half an ellipse. In the sand too, I have drawn half an ellipse.

What happened when I kept on going?

Beyond the Ellipse

Keeping going, the string wrapped around the stick. …

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