Set 2: Desargues' Configuration, Duality
2.1.
Construct two points X and Y, a line L on X and Y, and three
more points A, B and C on L. Hide the points X and Y and the line L.
Use any two of the set S = { A, B, C } to construct a segement.
Notice that by moving the third point, you can move it
onto the segment formed by the other two,
or you can move it so that
it is isolated and outside the segment you created, disconnected from the segment.
However, if you construct a second segment using a different pair from S,
(one point from the first pair will be in the second pair), you will
notice that after having done so, no matter how you
move A, B or C the points are never isolated, but are
always connected by what appears to be one segment. Experiment
with all permutations of positions of A, B and C to convince
yourself that this is so.
There is nothing to hand in for this part of the assignnment.
You will use what you have learned here in part 2.2.
2.2.
Recall that in lecture, the construction of
Desargues' configuration, was given (with somewhat different labels) as follows.
Start with
- any point V
- any three distinct lines x, y, and z on V
- any two distinct points A and A' on x and distinct from V
- any two distinct points B and B' on y also distinct from V
- any two distinct points C and C on z, distinct from V
Ensure that A' B', and C' are chosen so that they form
a proper triangle (that is, not collinear) .
So far we have built ABC and A'B'C'
to be
proper triangles that are in perspective from the point V.
We then constructed the three more X, Y and Z so that :
- BC and B'C' meet at X,
- CA and C'A' meet at Y and
- AB and A'B' meet at Z.
Finally we proved part of Desargues' Theorem by showing that
- the three points X, Y and Z are collinear.
Your job is to use GeoGebra to construct the dual of
this configuration. You will start with
- any line v
- any three distinct points X, Y and Z that lie on v
Notice that these two steps are the duals of the first
two steps listed above.
Also notice that we preserve the convention that
points are given by upper case letters, and lines are
named by lower case letters.
Complete the steps for the construction
of the dual figure, by dualizing each of the above steps.
Your finished figure will consist of ten points and ten
lines. Each point will be on three lines and each line
will be on three points.
As with question 2.1, include enough segments (20 in all?) so
that the appropriate portion of the line will be always be shown,
regardless of the sequential order of each collinear triple of
points.
Experiment with your figure by moving the line v,
the points X, Y and Z
on v, and any other controllable elements. Confirm
to your own satisfaction that things are working correctly.
Submit a
print of the figure and a listing of your Construction Protocol.
2.3.
This is a pen and paper exercise. For each of the following
Cartesian (Euclidean) problems, translate the question to one about
homogeneous coordinates in the projective plane. Solve the problem
in the projective plane using homogeneous coordinates, and then
translate the answer back to the Cartesian plane. Check your
answers in the Cartesian plane. Here are the questions:
- Find the line on the points whose Cartesian coordinates
are (3,7) and (4,9).
- Find the point of intersection of the lines whose equations are
| 3x+7y = -1 and -4x-9y = 1 |
- Find the intersection of x+8y = 0 and 3x+7y+1= 0.
- Find the line on the point (3,7) that has slope 8.
Hint: use the ideal point that contains all lines with slope 8.
Now that you have done these four exercises,
explain how "a." and "b." are duals of each other.
Do the same for "c." and "d.".
Last modified 2011-05-13