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
Replies to group please

On Wed, 23 Feb 2005 12:24:21 -0000, "John Rowland"
wrote:

"Larry Lard" wrote in message
oups.com...


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.

That's broadly correct, but there is a difference in that whereas
height is a continuous quantity which has a value for *every*
location, journey time from X only has a value at discrete locations,
namely those where there are stations. So if you want to construct a
contour map (or a carpet plot) on a map of (say) England then you need
to interpolate between the values using a straight line or a
mathematical function that does it more smoothly.

Here's an example: if it takes 40 minutes to West Drayton and 25
minutes to Slough (I'm guessing these times), then where does the "30
minute contour" run? The answer is it passes somewhere between the two
stations. The exact position depends on what type of interpolation you
are using.

Note that you don't only have to interpolate between adjacent stations
on the same line - you need to interpolate between adjacent stations
irrespective of whether there is a railway between them, to allow you
to work out, say, where the 30 min contour line passes between Slough
and Gerrard's Cross.

There is software around that will do all this and take a set of
(X.Y) point and plot the Z value using contours. There are three steps
we need to take:

* obtain the Grid Reference (X,Y) of every station in (say) the
south-east. Does this data already exist?
* look up the fastest peak journey time (Z) from somewhere to each
station
* plot the resulting data as a contour plot

PaulO

Stephen Osborn wrote:
John Rowland wrote:
"Stephen Osborn" wrote...

However the contours on an OS map (and AFAIK isobars on a weather chart)
never touch let alone cross.

They can touch, but they can't cross.

I think you are wrong there.

Contours mark places of equal height. If two contours touch at any one
point then, de definito, they have to touch at *all* points, so the two
contours become one contour.

You already have the counterexample of a vertical cliff. I have seen
those in nature - although not all of the cliff was vertical, there were
certainly parts that were, and they were definitely big and vertical
enough for contours to meet on the map.

Whereas, in the example that Mike gave,
the isochrones will have to cross.

No, they won't. It's just the same as a weather map, it's just a map where
every point has a real number associated with it. Draw two isochrones
crossing each other, write various times on the isochrones and on the spaces
between them, and you'll see that it can't happen.

I was accepting Mike's point that "I think it can't be done on a flat
map without rearranging the order of stations on each line."

Using Mike's example, a 'railway straight line' runs Wimbledon, Raynes
Park & Surbiton in that order. The isochrone passes through Wimbledon &
Surbiton (ignoring the 1 minute difference) but not through Raynes Park.

That arrangement is possible on an OS map or weather chart as, say, two
maxima (M) can be separated by a minimum (m) so there will be places
with the same value but they are not linked by a contour / isobar. For
example a1 & a2 in the diagram below:

a b a
a a b b a a

a M a1 b m b a2 M a

a a b b a a
a b a

Here the contours / isobars that a1 & a2 sit on have different centres.

For a travel map to be of use, every point on it has to share share the
same centre.

That statement is absolutely ridiculous! I'd go so far as to say the
converse is true: If they don't share the same centre then the map can
clearly display the information* and is therefore useful. If they do
share the same centre then it's impossible to display sufficient
information clearly, therefore the map would be useless.

They do have to share the same reference point, but that's all. I assume
what you were trying to say was that to be useful for determining the
time it would take to get between any two points on the map, every
contour has to share the same centre. If that is what you mean, I'm not
going to bother disputing it because it's pointless - software could do
the job a lot better than any map!


* The easiest way of doing so would be to set the background colour
according to how long it takes to get to the reference point. I expect
this would be referred to as isochromic isochrones!

Aidan Stanger wrote:
Stephen Osborn wrote:

John Rowland wrote:

"Stephen Osborn" wrote...


However the contours on an OS map (and AFAIK isobars on a weather chart)
never touch let alone cross.

They can touch, but they can't cross.

I think you are wrong there.

Contours mark places of equal height. If two contours touch at any one
point then, de definito, they have to touch at *all* points, so the two
contours become one contour.


You already have the counterexample of a vertical cliff. I have seen
those in nature - although not all of the cliff was vertical, there were
certainly parts that were, and they were definitely big and vertical
enough for contours to meet on the map.


Whereas, in the example that Mike gave,
the isochrones will have to cross.

No, they won't. It's just the same as a weather map, it's just a map where
every point has a real number associated with it. Draw two isochrones
crossing each other, write various times on the isochrones and on the spaces
between them, and you'll see that it can't happen.

I was accepting Mike's point that "I think it can't be done on a flat
map without rearranging the order of stations on each line."

Using Mike's example, a 'railway straight line' runs Wimbledon, Raynes
Park & Surbiton in that order. The isochrone passes through Wimbledon &
Surbiton (ignoring the 1 minute difference) but not through Raynes Park.

That arrangement is possible on an OS map or weather chart as, say, two
maxima (M) can be separated by a minimum (m) so there will be places
with the same value but they are not linked by a contour / isobar. For
example a1 & a2 in the diagram below:

a b a
a a b b a a

a M a1 b m b a2 M a

a a b b a a
a b a

Here the contours / isobars that a1 & a2 sit on have different centres.

For a travel map to be of use, every point on it has to share share the
same centre.


That statement is absolutely ridiculous! I'd go so far as to say the
converse is true: If they don't share the same centre then the map can
clearly display the information* and is therefore useful. If they do
share the same centre then it's impossible to display sufficient
information clearly, therefore the map would be useless.

They do have to share the same reference point, but that's all. I assume
what you were trying to say was that to be useful for determining the
time it would take to get between any two points on the map, every
contour has to share the same centre. If that is what you mean, I'm not
going to bother disputing it because it's pointless - software could do
the job a lot better than any map!


* The easiest way of doing so would be to set the background colour
according to how long it takes to get to the reference point. I expect
this would be referred to as isochromic isochrones!

That's the way the isochrones I've seen have done it. It makes for a
very clear and interesting picture. Comparing the difference in journey
times to two locations is also done this way (i.e. set the isochrones as
the difference in journey time, +/-, for reaching point B compared to
reaching point A).

--
Dave Arquati
Imperial College, SW7
www.alwaystouchout.com - Transport projects in London

Larry Lard wrote
[...]
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...

Yes. Thank you.

I really didn't do this on purpose.

The OP referred to distorted maps where travel time to a central point
was represented by distance.

John Rowland and others (reference to 'bubbles') were considering
overlays on standard geographical maps.

http://www.sciencenews.org/articles/20040828/bob8.asp and links
for explanation/discussion of weird maps, based on population and so
forth.

--
Mike D

In article ,
Paul Oter wrote:
Here's an example: if it takes 40 minutes to West Drayton and 25
minutes to Slough (I'm guessing these times), then where does the "30
minute contour" run?

A five minute walk from Slough Station.

Next?

--
Mike Bristow - really a very good driver

Dave Arquati wrote:
Aidan Stanger wrote:

Stephen Osborn wrote:

snip John Rowland wrote:

I think you are wrong there.

Contours mark places of equal height. If two contours touch at any one
point then, de definito, they have to touch at *all* points, so the two
contours become one contour.

You already have the counterexample of a vertical cliff. I have seen
those in nature - although not all of the cliff was vertical, there were
certainly parts that were, and they were definitely big and vertical
enough for contours to meet on the map.

I am still not convinced you will find a natural cliff that is *truly*
vertical (e.g. measured by a plumb line) for more than 10 metres, in
order to have the contour lines coincident.

I would be fascinated if you can think of an example where this is true.

For a travel map to be of use, every point on it has to share share the
same centre.


That statement is absolutely ridiculous!

Yes it is, isn't it. I had fallen into the same trap as Larry Lard (in
an earlier post) had and was thinking of the distance from the centre
showing the travelling time. Probably too much time looking at LU zonal
maps!

this would be referred to as isochromic isochrones!

Nice!

--

regards

Stephen

Stephen Osborn wrote:
I am still not convinced you will find a natural cliff that is *truly*
vertical (e.g. measured by a plumb line) for more than 10 metres, in
order to have the contour lines coincident.

I would be fascinated if you can think of an example where this is
true.

http://www.climbvertigo.ca/location-first-face.htm seems to indicate that in
places, the cliff is actually overhanging.

Tim (tm)
--
tim at economic-truth.co.uk Xbox Live gamertag: Xexyz
http://www.economic-truth.co.uk - the students' economics resource
http://www.ugvm.org.uk - the uk.games.video.misc magazine
The talkabout network is denied permission to reproduce this post

Tim Miller wrote:
Stephen Osborn wrote:

I am still not convinced you will find a natural cliff that is *truly*
vertical (e.g. measured by a plumb line) for more than 10 metres, in
order to have the contour lines coincident.

I would be fascinated if you can think of an example where this is
true.


http://www.climbvertigo.ca/location-first-face.htm seems to indicate that in
places, the cliff is actually overhanging.

Tim (tm)

Yes but that is not vertical so the contour lines (of different heights)
would not be coincident.

I am struggling to get my mind around how an overhang should be shown on
an OS map.

--

regards

Stephen

Stephen Osborn wrote:

http://www.climbvertigo.ca/location-first-face.htm seems to indicate
that in places, the cliff is actually overhanging.

Yes but that is not vertical so the contour lines (of different
heights) would not be coincident.

I am struggling to get my mind around how an overhang should be shown
on an OS map.

It would involve contour lines crossing. Each would have to be clearly
labelled at every point.

Tim (tm)
--
tim at economic-truth.co.uk Xbox Live gamertag: Xexyz
http://www.economic-truth.co.uk - the students' economics resource
http://www.ugvm.org.uk - the uk.games.video.misc magazine
The talkabout network is denied permission to reproduce this post