The
cross ratio of 4 collinear points (a pencil of points, or a range of
points) is given by;
The
cross ratio of 4 collinear points (a pencil of points, or a range of
points) is given by;
where 
etc
is the signed magnitude of the segment AC, so that if, for
this particular configuration,
is taken as positive 
is taken as negative.
Create some diagrams of pencils of points A ,B, C, D in various
positions, measure the lengths of AC etc, and evaluate 
.
Place the points so that the cross ratio is positive and negative,
very large (
),
very small (
).
Clearly there are 24 different orders of the cross ratio
etc. Calculate some of these. How are they related? How many
different values are there? (Note: this explains why authors choose
different definitions of the ratio of ratios.
Draw
a pencil of points A, B, C, D. Choose an arbitrary point P not on the
line, and draw the lines PA etc. Draw an arbitrary line across the
page and mark the intersections of this line with PA etc. Label the
points of this pencil A' etc.
Evaluate
and 
.
Change the arbitrary line creating A' etc, change the position of P.
Comment.
The
cross ratio of 4 concurrent lines (a pencil of lines) is given
by:
where 
is the oriented, signed angle between the lines a and c, so
that, for this particular configuration, if the angle between a and c
rotating anti-clockwise is taken as positive, then the angle between
d and b rotating clockwise is taken as negative.
Create some diagrams of pencils of lines a, b, c, d , measure the
angles and evaluate the cross ratio 
.
Change the orientations of a, b, c and d so that the cross ratio is
very large, positive, negative etc.
Draw
a pencil of lines, a, b, c, d and draw a line, label the
corresponding intersections A, B C, D. Evaluate
and .
Comment.
(The property to which this draws attention may be used to demonstrate the equality which can be noticed in Task 2. The property needs proving itself, of course.)
For a pencil of lines, what is the analogous task to Task 3?
If the points A and B are represented by position vectors
and 
then C and D of the pencil can be represented by:
and
.
Define the cross ratio using
this vector form as:
Draw a line, choose points A and B. Choose points C and D on AB so that they divide AB in nice ratios and so give nice numbers for the parameters m and n etc. Evaluate the vector form of the cross ratio and compare this with the evaluation of the signed segment definition. Make appropriate choices so that some of the parameters are negative.
(What happens if you choose to take vectors such as 
for any numbers m and n? Where are C and D? Does the cross ratio
definition still make sense, given that A,B,C,D no longer form a
pencil of points?)
If
the points A ,B, C, D are represented by the complex numbers a, b, c,
d, then the cross ratio is given by:
Use the complex number definition to evaluate the cross ratio for
a pencil of 4 points relative to some coordinate axis.
Compare this value with that obtained using definition 1.
Impose a different coordinate axis on the same points and evaluate the new cross ratio.
Choose
4 points which are concyclic and use the complex number definition to
evaluate the cross ratio.
Choose another point ,V,on the same circle and evaluate the cross
ratio for the pencil of lines VA, VB etc.
Use the magnitudes of the segments AC etc for these concyclic
points to evaluate
(How are you going to interpret signed magnitudes?)
What happens to the value of the cross ratio if the points are neither collinear nor concyclic?
