hmmm - I'm not a math expert by any means, but I
thought there would be some sort of convergence
here...the "midline" has to lie somewhere, and it is
possible for there to be a finite and accurate midline
between two infinite lines. Given that the midline
would be a function over a finite distance, I think
the length of the midline would be finite also.
--- "Brian J. Butler" <
bjbutler@...>
wrote:
> On Wednesday 16 April 2003 07:12 pm, you wrote:
> The thalweg is also of indefinite length because
> there can be an infinite
> number of changes in the y value over any interval
> of x.
> BJB
>
> > The riverbanks can have an almost infinite length
> > which is practically impossible to measure, but I
> > think the center or thalweg can not be considered
> in
> > the same category. Looking at it from above with
> one
> > bank on the bottom and the other on top (thus
> > traveling along the bank is traveling along the
> x-axis
> > and crossing river is along y-axis), there emerges
> an
> > upper and lower bound. When drawing the boundary
> > along the center or thalweg, you can only go in
> one
> > direction without doubling back and must stay
> within
> > the bounds...essentially only one y-coordinate per
> > each x-coord. This is unlike the bank which, due
> to
> > shapes of rocks, you can have the shore go in and
> out
> > (several y-coord per x-coord).
> >
> > --- "Brian J. Butler" <bjbutler@...>
> >
> > wrote:
> > > On Wednesday 16 April 2003 06:20 pm, you wrote:
> > > Why would the center of a river be any smoother
> than
> > > banks from which it is
> > > equidistant? And molecules, grains of sand,
> > > boulders, and great bends are all
> > > features that determine the edge of a river,
> hence
> > > its length. For some
> > > discussions it might be OK to ignore features
> below
> > > a certain size, but we
> > > are specifically discussing the length of the
> > > boundary, which is dependent on
> > > the scale of objects we consider.
> > > BJB
> > >
> > > > The center of a river is about as smooth as
> you
> > >
> > > can get.
> > >
> > > > Why would you measure a boundary line around a
> > >
> > > grain of sand?
> > >
> > > > Theoretically, this discussion could get into
> such
> > >
> > > things, but practically,
> > >
> > > > the line would go right across the top of that
> > >
> > > grain, or boulder, or rock,
> > >
> > > > not around it.
> > > >
> > > > -----Original Message-----
> > > > From: Brian J. Butler
> > >
> > > [mailto:bjbutler@...]
> > >
> > > > Sent: Wednesday, April 16, 2003 3:44 PM
> > > > To: BoundaryPoint@yahoogroups.com
> > > > Subject: Re: [BoundaryPoint] Re: American
> State
> > >
> > > Boundaries
> > >
> > > > On Wednesday 16 April 2003 05:12 pm, you
> wrote:
> > > > You are not on the right wavelength yet. The
> > >
> > > natural boundaries you
> > >
> > > > enumerated are not smooth curves that can be
> > >
> > > measured in the traditional
> > >
> > > > sense. I agree that you can determine a
> minimum
> > >
> > > length of these boundaries
> > >
> > > > by interpolating between fixed points on the
> > >
> > > boundary. But the true length
> > >
> > > > of the boundary depends on how small your
> samples
> > >
> > > are. For example, you
> > >
> > > > would have a longer measurement if you
> measured
> > >
> > > around each rock along the
> > >
> > > > riverbank, or each grain of sand. So you are
> > >
> > > doubly correct - your
> > >
> > > > estimate
> > > >
> > > > could be off by a great margin, an infinite
> margin
> > >
> > > perhaps, and the minimum
> > >
> > > > length of the OK-TX boundary is longer than
> the
> > >
> > > CA-NV boundary. I don't
> > >
> > > > think you can make the statement that the
> OK-TX
> > >
> > > boundary is longer than the
> > >
> > > > VA-WV boundary, though, for example, because
> it
> > >
> > > depends on how irregular
> > >
> > > > the
> > > >
> > > > boundaries are and how carefully you measure
> them.
> > > >
> > > > BJB
> > > >
> > > > > Well, *anything* has a length depending on
> how
> > >
> > > you measure it. But most
> > >
> > > > > US state boundaries have specific
> definitions
> > >
> > > that are actual places on
> > >
> > > > > the ground, whether it's mean highwater,
> center
> > >
> > > of channel. top of the
> > >
> > > > > ridge, etc. E.g., the Kentucky boundary
> along
> > >
> > > the Ohio River is the
> > >
> > > > > waterline on the northern bank, so KY
> controls
> > >
> > > the river. The definition
> > >
> > > > > IIRC was fixed in time so that it doesn't
> change
> > >
> > > as the Ohio rises or
> > >
> > > > > falls or carves new segments of the channel.
> > > > >
> > > > > Anyway, I am still curious if there is a
> place
> > >
> > > to ascertain the actual
> > >
> > > > > length of the OK-TX boundary? I estikated it
> as
> > >
> > > somewhere around 800
> > >
> > > > miles,
> > > >
> > > > > while CA-NV was just over 600 miles. My
> > >
> > > estimates could be off by a great
> > >
> > > > > margin, but I don't think they are off so
> much
> > >
> > > as to change the ranking.
> > >
> > > > > -----Original Message-----
> > > > > From: Brian J. Butler
> > >
> > > [mailto:bjbutler@...]
> > >
> > > > > Sent: Wednesday, April 16, 2003 12:12 PM
> > > > > To: BoundaryPoint@yahoogroups.com
> > > > > Subject: Re: [BoundaryPoint] Re: American
> State
> > >
> > > Boundaries
> > >
> > > > > On Wednesday 16 April 2003 01:18 pm, you
> wrote:
> > > > > A natural boundary, such as a river, has a
> > >
> > > length that depends on how
> > >
> > > > > closely
> > > > > you measure it.
> > > > > BJB
> > > > >
> > > > > > But CA-NV wouldn't be the longest border
> > >
> > > between two states, straight
> > >
> > > > > > or not. The OK-TX border for a good
> distance
> > >
> > > is the meandering Red
> > >
> > > > > > River. There's no basis to say that
> doesn't
> > >
> > > count as distance and that
> > >
> > > > > > one should draw an imaginary "straight"
> line
> > >
> > > instead to cut the
> > >
> > > > > > corners. The boundary is the boundary line
> > >
> > > itself.
> > >
> > > > > > What is the length of the CA-NV boundary
> and
> > >
> > > the OK-TX boundary?
> > >
> > > > > > -----Original Message-----
> > > > > > From: Brian J. Butler
> > >
> > > [mailto:bjbutler@...]
> > >
> > > > > > Sent: Wednesday, April 16, 2003 9:39 AM
> > > > > > To: BoundaryPoint@yahoogroups.com
> > > > > > Subject: Re: [BoundaryPoint] Re: American
> > >
> > > State Boundaries
> > >
> > > > > > On Wednesday 16 April 2003 11:19 am, you
> > >
> > > wrote:
> > > > > > CANV is certainly the straight-line champ.
> If
> > >
> > > we "go fractal" maybe
> > >
> > > > > > ID-MT, OK-TX, or even VA-WV would take the
> > >
> > > cake.
> > >
> > > > > > BJB
> > > > > >
> > > > > > > nice question & nice answer
> > > > > > >
> > > > > > > how about canv for longest
> > > > > > >
> > > > > > > --- In BoundaryPoint@yahoogroups.com,
> Brian
> > >
> > > J. Butler
> > >
> > > > > > > <bjbutler@b...> wrote:
> > > > > > > > On Wednesday 16 April 2003 09:50 am,
> you
> > >
> > > wrote:
> > > > > > > > The shortest is easy - at AZ-CO-NM-UT
> > >
> > > there are two pairs of
> > >
> > > > > > > states that meet
> > > > > > >
> > > > > > > > at a point.
> > > > > > > > BJB
> > > > > > > >
> > > > > > > > > Which state shares the longest
> border
> > >
> > > with another state?
> > >
> > > > > > > (The border
> > > > > > >
> > > > > > > > > does not have to be continuous.)
> Which
> > >
> > > state shares the
> > >
> > > > > > > shortest?
> > > > > > >
> > > > > > > > > Your use of Yahoo! Groups is subject
> to
> > > > > > >
> > > > > > > http://docs.yahoo.com/info/terms/
> > > > > > >
> > > > > > > > --
> > > > > > > > Brian J. Butler
> > > > > > > > BJB Software, Inc.
> > > > > > > > 508-429-1441
> > > > > > > > bjbutler@b...
> > > > > > > > http://www.bjbsoftware.com
> > > > > > >
> > > > > > > Your use of Yahoo! Groups is subject to
> > > > >
> > > > > http://docs.yahoo.com/info/terms/
> > >
> > > --
> > > Brian J. Butler
> > > BJB Software, Inc.
> > > 508-429-1441
> > > bjbutler@...
> > > http://www.bjbsoftware.com
> >
> > __________________________________________________
> > Do you Yahoo!?
> > The New Yahoo! Search - Faster. Easier. Bingo
> > http://search.yahoo.com
> >
> >
> >
> >
> > Your use of Yahoo! Groups is subject to
> http://docs.yahoo.com/info/terms/
>
> --
> Brian J. Butler
> BJB Software, Inc.
> 508-429-1441
> bjbutler@...
> http://www.bjbsoftware.com
>
__________________________________________________
Do you Yahoo!?
The New Yahoo! Search - Faster. Easier. Bingo
http://search.yahoo.com