John Rowland wrote:
"Larry Lard" wrote in message
ups.com...

John Rowland wrote:
"Larry Lard" wrote in message
oups.com...

There are many "saddle" points in the landscape where,
say, the land is lower to the north and south, and higher
to the east and west. The contour which marks the height
of the saddle point runs away from the saddle point
in 4 directions. It would be perverse to describe the
contour as crossing itself, but the contour could
meaningfully be described as touching itself
at this one point. This is as true for a map of
isochrones or isobars.

Like this, yes? :

[fixed width font needed]
numbers are heights

0 -1 -2 -3 -2 -1 0
1 0 -1 -2 -1 0 1
2 1 0 -1 0 1 2
3 2 1 0 1 2 3
2 1 0 -1 0 1 2
1 0 -1 -2 -1 0 1
0 -1 -2 -3 -2 -1 0

Surely in this situation there is only one contour, though, and it
is
X-shaped. If you want to argue that there are two contours meeting
in
the middle, how do you decide whether it's a meeting a , or a ^
meeting a v ?

It's a ^ meeting a v. Look at the bigger picture, which would be like
this...

-------
--0-0--
-0+0+0-
--0-0--
-------

I call this one 8-shaped contour, rather than two O-shaped contours
that touch at one point, but I would invest about zero effort in
arguing the toss.



Anyway, I'm not really sure this branch of this thread (?) has
anything
useful to say about the original problem, as raised by Michael
Dolbear:
"I think it can't be done on a flat map without rearranging the
order
of stations on each line."

His statement is so clearly wrong it's hard to argue with it until
someone
explains why they think it's right. Every public point in the 2D
space has a
scalar quantity associated with it, namely journey time from point X.
Mathematically this is identical to the contour maps, where every
point
which is not inside a building has a scalar quantity associated with
it,
namely height above sea level.

OK yes here I have been incorrectly thinking about something completely
other.

The 'it' you can't do without rearranging stations: make such a map
where commuting time is represented by distance;

What you *can* do: make such a map where isochrones are explicitly
overlaid on a geographical map, obviating any need to rearrange
stations.

At some point the 'it' changed and I failed to notice...

--
Larry Lard
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