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Painting by Functions

Age 16 to 18
Challenge Level

Why do this problem?

This problem gives an engaging opportunity for genuine cross-curricular work whilst bringing in ideas from composition and transformation of functions. It forms the basis of the ideas underlying Computer Generated Images which are of fundamental importance in the gaming industry, into which many students might go.

Possible approach

Find some images of interest and discuss ways in which they might be mathematised.

There are three levels of possibility:

1) Look at the images and discuss how they might by represented by standard mathematical shapes.
2) Decide how the outlines of key regions of the images might be represented by mathematical curves.
3) Use graph paper or a graphing package to start to quantify precisely the shapes or curves: the goal is explicitly to find equations corresponding to the key parts of the image.

You might suggest that students are only allowed, say, 7 shapes or 7 curves with which to represent the image. This will help to focus on the key aspects of the images: the goal is to create a simple, abstract rendering of the image.

You might also suggest that students prepare the same images with their choices of abstract curves and then ask someone from the art faculty to determine which best represents the images.

Key questions

What are the key aspects of the image?

Can you see any lines which look like part of a standard function?

What is the equation of a line / ellipse / parabola?

How do you make a curve move left/right or up/down?

How do you stretch or squash a curve?

Possible extension

Students can take this task as far as they wish, in both the artistic or mathematical directions. Suggestions for further reading are given at the foot of the main problem.

Possible support

You could print off the images and provide transparencies onto which the students could draw as preparatory work.

If the equations of curves' shapes are causing a problem, you might want to start with straight lines and the problem Painting Between The Lines.