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An article demonstrating mathematically how various physical
modelling assumptions affect the solution to the seemingly simple
problem of the projectile.
Published July 2005,August 2005,February 2011.
Unfortunately, the first model that springs to mind may not be
the most appropriate. In this case it is tempting to think first of
Hooke's law of elasticity. This is after all a very well known
mathematical model of how things like springs and elastic bands
behave, giving a pretty accurate description of how the forces
change as you stretch a spring. Paraphrased, it says that the force
is proportional to the distance stretched.
Unfortunately, Hooke's law in this context is a very good
example of a completely inappropriate mathematical model. In the
virtual geoboard, you can't feel any forces through your mouse as
you manipulate the rubber bands. We have to look elsewhere for a
way to create our illusion..
To get to a more useful model I needed to think about how the
user would create and manipulate those rubber bands. Some of this
was straightforward user interface design. I would represent a pot
of rubber bands with a button. Pressing the button - I called it
the 'BandFactory' in my code - would create a small relaxed band
that could be dragged on top of a peg. Pressing the mouse button
again over this band and dragging would stretch the band, and if I
released the mouse button when I was over another peg, then the
band would attach to the new peg.
I would then have something like this:
From this point it would be natural to insert my finger (really
the mouse pointer) inside the rubber band, and drag it over another
peg. If I dragged down a little, then I'd expect to see something
like this: (The small purple circle is representing my finger
I might also drag upwards, in which case I'd expect a result
So far so good, and I haven't needed to use very much
mathematics. Or have I? Well, the truth of the matter is that the
first time I tested this part of the program, the result was not
what I wanted. It sometimes did the right thing, but sometimes -
depending on where the two original points were - I would see
This wasn't as surprising as it might seem partly because I
never expect things to work first time and partly because I knew
that I'd left a little bit of code to sort out later. I hadn't yet
given the machine a good way to decide which arm of the band needed
to be moved. In this case, dragging up should move the upper arm,
and dragging down should move the lower arm. When the program went
wrong it was choosing the wrong arm.
I'm also glossing over the coordinate geometry that was needed
to draw the rubber bands going round the pegs. Suffice to say that
the algorithm used to draw the rubber band going round a peg also
ensured that it developed an extra loop around my finger whenever
the program misbehaved and picked the wrong arm to drag.
In the internal numerical world of the machine, this problem
looks different. It changes character and becomes algebraic. Let's
try to recast the problem in these terms.
Here, $A$ is a point with coordinates $(a_0, a_1)$, and $B$ is
at $(b_0,b_1)$. The program draws the elastic band by choosing some
starting point a distance $r$ from the line segment $A B$. It then
circumnavigates the line segment $A B$ always maintaining that
fixed distance $r$ and drawing a purple line as it goes. The two
arms are distinguished by direction - one is the arm drawn from $A$
to $B$, and the other goes from $B$ to $A$. A third point - my
finger at $C$ - is then introduced and dragged to $(c_0, c_1)$.
We are now in a position to restate the problem. Given $A=(a_0,
a_1)$, $B=(b_0,b_1)$ and the moving $C=(c_0, c_1)$, how is the
machine to know whether to insert $C$ in the arm $AB$ or the arm
$BA$? Assume a circumnavigation of $AB$ in the positive
anticlockwise sense. The problem does not look so easy now, but it
is immediately apparent that if the circumnavigation was in the
opposite sense, $C$ would insert in the opposite arm.
I'll leave that one with you for a bit while we look at another
little problem that arises from our desire to mimic rubber band
behaviour. This second problem is, "How do we get the bands off the
pegs?" and it turns out to be closely related to our first problem.
Look at how the situation plays out in the next few diagrams:
Here, I had previously made a parallelogram by wrapping the
points $A$, $B$, $C$ and $D$, and I'm about to unhook the band from
$A$. You can see that the band is still attached to peg $A$ since
it changes direction there. If I pull down just a little further,
$A$ would be wholly inside the band and unattached:
If I now let go of the mouse button to simulate removing my
finger, then the band should snap to wrap around $B$, $C$ and $D$
only in a rather pleasing elasticated way. (Note to self: I really
must add some sound effects!)
Aha! Things are beginning to make sense. The rubber band
detaches itself from $A$ because it refuses to bend inwards to stay
attached. If we constrain the rubber bands so they always make
convex shapes we have a natural way to mimic in 2D the 3D act of
lifting a band off a peg.
This is a really helpful observation because I know that this
problem has been solved before. It's the problem of finding the
smallest convex hull that encloses a set of points. I type 'convex
hull' into Google, and find everything I need to know on the
Convex Hull Algorithm ' - courtesy of an excellent Australian
web site. There is a bonus in the shape of one component of this
algorithm known as the 'can see test'.
The core of the incremental convex hull algorithm is a small
test needed to determine whether the points of a triangle are
labelled in a positive anticlockwise sense, or a negative clockwise
sense. I hope this is beginning to sound familiar to you - and yes
- it will also provide the solution to our earlier problem. The web
page suggests that we need to worry about calculating the area of
the triangle and see whether it comes out positive or negative.
This might seem like a strange idea - that areas can be negative -
but it arises quite naturally if we attempt to calculate areas
using coordinate geometry.
The test is known as the 'can see' test because one imagines
that only one side of each line is visible to another point. The
test determines whether a line presents its visible aspect to the
Let's go back to our orginal problem involving the fixed points
$A$ and $B$, and the movable point $C$ and try to calculate the
area of the triangle $A B C$. The virtual geoboard itself is quite
a useful tool in doing this. (I knew there was a reason to go
through all this!)
The machine only knows about coordinates, so it's quite useful
to surround the triangle with the smallest enclosing rectangle with
sides parallel to the axes. On the geoboard it looks like this
(I've clicked on one of the vacant green pegs to minimise their
I said 'rectangle' because in general the bounding box of a
triangle will be rectangular rather than square. I'll continue to
use the term 'rectangle' in the text as this is the general case
even though in my diagrams above and below many of the rectangles I
am talking about happen to be square.
Let's add a few more bands to help calculate areas:
Notice that triangle $B C E_1$ is half of the green rectangle,
triangle $C E_3 D_1$ is half of rectangle $C E_3 D_1 E_1$, and
triangle $B E_1 D_2$ is half of rectangle $B E_1 D_2 E_2$.
Furthermore, we can transform the two triangles $B E_1 D_2$ and $C
D_1 E_1$ to areas inside $A B C$ by moving $D_1$ and $D_2$ in area
and so demonstrate that the area of triangle $A B C$ is one half
the sum of the areas of the 3 green bands below. $A$ is the special
point of the triangle because it is the one point that coincides
with a corner of the triangle's bounding box. There must always be
at least one such point.
The area can now be calculated easily in terms of the
coordinates $a_0, a_1, b_0, b_1, c_0, c_1$. I won't spoil the fun
by doing this for you, but you should be able to verify that the
resulting expression is either positive or negative depending on
whether $A$, $B$, and $C$ cycle in the positive or negative sense.
When the area is zero, the points $A$, $B$ and $C$ are colinear,
and of course in that situation we don't need to decide between
moving $AB$ or $BA$ since we move neither.
Putting everything together, we now have a model that can draw
and manipulate convex polygons. Should we go further, and implement
concave polygons too? It is after all possible to create a concave
polygon on a real geoboard by lifting the band over selected
Could we do this and still use the convex hull technique? Yes -
I think we could. Things would get a little more complicated. The
program would have to record whether pegs used in a polygon were
attached on the inside or on the outside, and I would have to
modify the drawing alogorithm so it weaved in and out as needed. I
would also need to be able to select a band to manipulate so my
'finger' could start on the outside and 'know' which band to lift
over pegs. This is all possible, but begins to violate the
engineers 'KISS = Keep It Simple, Stupid' maxim. So I'm reluctant
to allow concave polygons on these pragmatic grounds.
Fortunately, there's a much stronger reason for rejecting
concave polygons - and it's a mathematical one. Convex shapes
created with rubber bands have the very useful property that the
angles between the straight line segments are invariant when we
change the peg size. Angle invariance is violated by concave
polygons. Since one of the main purposes of using the geoboard is
to investigate angles, this is a property we cannot afford to lose.
Our virtual geoboard is in fact better than the real one because it
excludes the possibility of creating a concave polygon with a
single rubber band.
Concave polygons can still be created with the virtual geoboard.
You simply have to use more than one rubber band to make them. You
could, for example, use one band for each line segment. Angles
would still be preserved when peg sizes change.
We men are prone to this. I'm sure you've seen us do a job and
then stand back and admire it for half an hour. I have to admit
that this was one of those jobs for me. I find it strangely
beautiful that a rather obscure algorithm is responsible for
something so tactile as making it feel like you are manipulating a
stretchy rubber band when you are actually just clicking and
dragging a mouse. Hope you feel the same too!