[Maths-Education] review of interactive geometry software

[Alexandre Borovik]

Dear Kate,

I personally find it useful to try to understand the nature and 
limitations of the mathematical engine at the heart of a particular 
maths learning package. The following simple experiment with Cinderella 
and Geogebra is a fragment of my paper at ALT News Online, 
http://newsletter.alt.ac.uk/4edkkzb138s
(a more complete version is available at
http://manchester.academia.edu/documents/0138/7244/Mathematics_and_IT.pdf):

"It is interesting to compare the behaviour, in Cinderella and GeoGebra, 
of a simple interactive diagram: two intersecting circles of varying 
radii and the straight line determined by their points of intersection. 
In GeoGebra, when you vary the radii or move the centres of the circles 
and make the circles non-intersecting, the line through the points of 
intersection disappears -- exactly as one should expect. In Cinderella, 
the line does not disappear, it moves following the movements of the 
circles, always separating them; when circles touch each other and start 
to intersect again, the line turns to be, again, the common tangent line 
of two circles or, in the case of two intersecting circles, the line 
through the points of intersection. (The line is called the radical axis 
of the two circles.)

To a mathematician, the behaviour of this diagram suggests that the 
underlying mathematical structure of GeoGebra is the real Euclidean 
plane. In Cinderella, the underlying structure is the complex projective 
plane; what we see on the screen is just its tiny fragment, a real 
affine part. The radical axis of two non-intersecting circles is the 
real part of the complex line through two complex points of 
intersection. The intersection points of two real circles are complex 
conjugate, the line is invariant under complex conjugation and therefore 
is real and shows up on the real Euclidean plane. For a mathematician, 
this is a strong hint that Cinderella could work better than GeoGebra in 
accommodating non-Euclidean geometries: elliptic and  hyperbolic  (the 
Lobachevsky plane) since they happily live in the complex projective plane."

Best wishes -- Alexandre

On 11/08/2010 18:02, Kate Mackrell wrote:

> I am currently writing a comparative review of
> Sketchpad/Cabri/Cinderella/Geogebra and would love to hear from anyone
> on the list concerning:
>
> a. their reasons for choosing to use a particular interactive geometry
> software. Don't necessarily limit yourself to the ones above - I would
> possibly be open to including other softwares in the review.
>
> b. any geometry/algebra tasks that would provide a useful basis of
> comparison for the softwares. I have a number of ideas, but I'm worried
> that I will automatically choose a task that works best with my own
> favourite software, which would not be quite fair!
>
> Thanks
>
> Kate Mackrell