Re: Contour maps, etc.

[LUDWIK KOWALSKI <kowalskil@alpha.montclair.edu>]

On Sat, 3 May 1997 John P. Ertel gave us references on contour plotting.
Thanks for this information. He also commented on the example which was
used to fomulate the problem:

> Sorry, the contour plot for the data that you have listed is a single 
> flat surface at Z=10 --- the iso-Z curve is completely undetermined.

I did not express my concern clearly. Let me do this again. Creating a
contour plot map is trivial (I wrote my own program for this yesterday)
when a complete set of data is available, that is when a value of Z is 
assigned to every possible x,y location. These locations are defined by 
a mesh grid chosen for the purpose. Why is it trivial? Because we know
that contour lines are continuous and non-crossing. All is simple when
a function Z(x,y) is specified by a formula. In that case we can generate 
as much data as we need at any level of accuracy (in principle).

But, if I am correct, the experimental data on Z(x,y) are often limited
and many mesh points must be filled by interpolations. This may become
a source of errors. Three-dimensional interpolations are not as simple 
as in two dimensions. Even linear interpolations are not well defined. To 
interpolate we must build "roof sections" above the x,y plane. Any three 
"columns" (x1,y1; x2,y2 and x3,y3) will define a roof element (for a local 
linear interpolation) but there are many ways of choosing triangles from 
the filled data bins on the x,y mesh grid. Should the "column sets" be 
selected on the basis of proximity or on the basis of flatness? And how is
this influenced by the sets which were already chosen and used? The order 
in which we proceed may become important.

An ideal contour "line" should have a variable width representing "error
bars" in consecutive locations. And distances between the equal-Z lines 
must be chosen to prevent overlappings. A contour map with "partially
overlapping lines" can lead to unreliable predictions. How many maps 
created by software packages would show such overlappings if the initial 
data were processed by an ideal program?
                                                        Ludwik Kowalski
Here is my example again:

One contour could be 1-2-3-4-5-6-7-8-1, another could be 1-2-3-8-4-5-6-7-1,
>    x    y    Z                      etc. These contour lines are acceptable
>   -------------     1         7     because they can be linked without
>    1    1   10           8          crossing. The contour Z=30 may be  
>    2    1   10      2         6     somewhere inside, and the contour
>    3    1   10                      Z=10 may be somewhere outside, of
>    3    2   10      3    4    5     what is shown and implied. The data 
>    3    3   10                      are MIXED WITH EACH OTHER randomly,
>    2  2.6   10                      for example with 2.8,2.9,30 or
>    1    3   10                      0.5,0.3,10  etc. (not shown here).
>    1    2   10                      The question was "how are the non-
                                      adjacent data points linked by a 
                                      computer program to create contours"?