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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.