August 24, 2005
The next part of solving the
probability problem I set out a week or so ago is proving to be
cantankerously obstreperous. The problem, to be sure, is,
given a random set of 3 vertices in a unit square, what are the
areas of shadow polygons, by which I mean the polygonal regions
outside the triangle where a fourth vertex would not complete a
convex quadrilateral.
I thought mebbe one could use
information about how the edge points are distributed around the
perimeter of the unit square to determine what sort of polygonal
shadows occur, but, as the diagram here show, that doesn't seem
likely.
The numbers below each square indicate the distribution
of edge vertices, starting with the left side and going
around clockwise. This is only a small sampling of the
possibilities, of course, but it shows some of the potential
complexity. Notice that the two 3300 entries have quite
different shadow polygons: the first has two triangles and a
quadrilateral and the other, two triangles and a hexagon.
How about a simpler subproblem? How many ways are
there to draw two line segments intersecting on the interior
of a square? Two configurations will only be
considered different if one cannot be obtained from the
other by 


Sliding vertices along an
edge (but not to a new edge.)

Rotating and/or flipping the
square.
There are 5 different ways that I can find
(shown at right.) This is equivalent to saying how
many ways are there to write 4 numbers, each 0, 1 or 2,
whose sum is 4. Two such sequences are different only
if one can't be obtained from the other by "rotating" or
reversing the sequence.
We can list all 9 possible arrangements of edge points
around the perimeter like so:
3
 3 
3 
3 
3 
3 
2 
2 
2
0 3  1 2 
1 1  0 2
 0 1  0
0  2 2 
1 2  1 1
0  0
 1 
1  2
 3 
0 
1 
2
For each we'd like to know how many different shadow
polygon outcomes are possible. For instance, for the
first (3300) we have two, as described above. For
3210, as shown below, there are either 3gon, 4gon,
4gon (red) shadowgons, or 3gon, 3gon and 5gon shadowgons.
When compared with 3300 pair, I have a way of saying
which is which: is the intersection of 1st and 3rd
edges(counting x1, x2, x3 from left to right) above or below
the line emanating from the 2nd left edgepoint? 

3120 is obtained from
3210 by moving the rightmost vertex on the top edge further
right and down around the upper right corner, so it's situation is
pretty much the same as 3210.
Truth be told, after making all
these diagrams, I'm pretty tired of the nitty gritty. Maybe
I'll come back to this with a fresh mind later on. For now,
the tedium of this makes me think moving on to parsing would be a
good thing.
