Geared Up: Exploring curves with Autograph

Robert Ward-Penny and Douglas Butler describe using the software package Autograph with a group of mathematics PGCE students at the University of Warwick.

Discovery and manipulation

One of the many benefits of using an ICT package such as Autograph to explore graphs in mathematics is the fact that learners get instant, visual feedback. Graphical outputs have a quality different from discrete ‘right’ or ‘wrong’ comments as they can suggest to the learner how close they are to achieving their intended goal, and allow them to work in a step-by-step manner towards a solution. In this way the software package can enable pattern spotting, generalisation, and even the occasional serendipitous discovery. It also opens the door to the discovery and manipulation of some unusual families of curves.

Recently we conducted an activity with the mathematics PGCE students at the University of Warwick which focused on two such sets of curves. The exercise was particularly successful, with the students producing some fantastic pieces of work. At each stage Autograph enabled the students to get to grips with the underlying mathematics in a way that would have otherwise been impossible.

Gear curves

The first set of curves was the gear curves. This is a particularly unusual set of curves which closely resemble gears.

Further details can be found at MathWorld here.

One form of gear curve can be drawn using the parametric equations:

x = (1 + 0.1 tanh (10 sin nt)) cos t, y = (1 + 0.1 tanh (10 sin nt)) sin t

where n is a constant and t is a parameter ranging from 0 to 2π radians. The picture below shows this curve when n=5.

A gear curve where n = 5

If you have Internet Explorer, Firefox or Safari browsers, you will be able to download the Autograph Player plug-in to explore this file for yourself. Unfortunately this facility is not yet available to MAC users.

Or if you have Autograph itself installed, you can download an Autograph file which works in Autograph.

You can get a 30 day trial version of Autograph here...

The most obvious starting point for the PGCE students’ investigation was to vary the parameter n and see what happened. In actual fact, n is equal to the number of teeth on the gear, and the students were soon experimenting with gears having hundreds of teeth. The next challenge was to create two interlocking gears. This required students to recognise how the Cartesian function transformations taught for GCSE mathematics could be extended to parametric equations. The third challenge was to devise a way to enlarge the curve, and to make three gears of different sizes that were all touching. The opportunities afforded by Autograph meant that all students were able to construct impressive looking systems, including those who were less experienced with using parametric equations.

Naturally, many students went above and beyond our expectations, and some devised a way to animate the gears. This in turn gave rise to further problems that needed to be overcome; how could a single animated parameter be used to turn some gears clockwise, and others anticlockwise? A fantastic piece of work produced by Phil Crockford, a former PGCE student, is offered here as an example of how one student tackled this challenge.

Cool cogs

Or if you have Autograph itself installed, you can download an Autograph file which works in Autograph.

You can get a 30 day trial version of Autograph here...

To get the gears to rotate anti-clockwise Phil has replaced the final t variable in each equation with t+kπ, then animated the graph setting 0≤k≤2. Using t - kπ instead will change the direction of rotation. As differently sized gears will need to rotate at different speeds, bigger or smaller gears require an additional scale factor to be placed before the k.

The gear curve expressions can be a little intimidating for learners who have not yet met the hyperbolic tangent function, so it can be valid to introduce them to learners as a ‘black box’. However, considering each component of the expressions can help enquiring learners understand the underlying mathematics better: for instance, the tanh function limits part of the equation to values between -1 and 1, and the sin function makes the output periodic. The values of 0.1 and 10 are related to the sharpness and relative amplitude of the teeth respectively. Once again, plotting all of these elements and animating constants can help to make this clear, and suitable choices of values can turn the gears into ‘windmills’ or ‘flowers’.

Rose curves

The second curve family that we explored with the PGCE students was the rose curves. These are a family of curves that were named by the Italian mathematician Guido Grandi in the 18th Century. The polar equation of these curves is r = cos(k?), where k is a constant.

The value of k is particularly critical for this family of curves, as different types of number lead to different types of curve. When k is an odd positive integer, the ‘flower’ formed has k petals. However, when k is an even positive integer, the ‘flower’ has 2k petals. Particularly interesting things happen when k is a fraction, or when k is irrational.

Rose curve where k = π

Or if you have Autograph itself installed, you can download an Autograph file which works in Autograph.

You can get a 30 day trial version of Autograph here...

The rose curves can give rise to some interesting pattern spotting exercises, and the slow plot function of Autograph can help learners recognise why these patterns come about. Graphical exploration can also give rise to questions which require a more algebraic approach; for example, what happens to the area of the rose curve as k changes? In fact, the area has one fixed value whenever k is an odd integer, and another fixed value when k is even; it is left as an exercise for the reader to determine these values!

In each of these cases, many students developed their understanding of the coordinate system being used through their explorations. The use of the graphing package also allowed students to meet curves beyond the standard set of ‘quadratic, cubic and reciprocal’. There are many other similar activities that could take place; you might like to get your students to explore the Lissajous curves using parametric equations, or challenge them to create a polar curve that resembles a particular shape, such as a heart. The potential for play is an important quality here, and it is one that can help learners of all ages become geared up for some motivating mathematics.