"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--
-------

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.

--
John Rowland - Spamtrapped
Transport Plans for the London Area, updated 2001
http://www.geocities.com/Athens/Acro...69/tpftla.html
A man's vehicle is a symbol of his manhood.
That's why my vehicle's the Piccadilly Line -
It's the size of a county and it comes every two and a half minutes